The notion that Orthodoxy rejects "Scholasticism" as such seems to me to come more from 20th Century anti-Catholic polemics
It was not until very late in the 19th century that scholastic philosophy became a part of Roman Catholic dogma (1871/Vatican I). Therefore it is *not at all* surprising that Orthodox focus on the rejection of scholasticism as ais found especially after it appears enshrined dogmatically in Vatican I. So also Aquinas's philosophy was controversial among Roman Catholics from the period of Aquinas until Vatican I, as Roman Catholic philosopher Frederick Copleston observed in his book on Aquinas. So "Orthodox objection" is not an isolated case; we can speak of Roman Catholics rejecting Aquinas before Vatican I in the 1870s as well (we will also further consider its rejection among many contemporary philosophers including an evident majority of self-described Roman Catholic philosophers and others below).
There is also the matter of rejecting elements of Aquinas' philosophy not simply qua philosophy, but as being incorporated as a dogma of the Christian faith. We see nothing resembling such a bold and innovative claim in the entire first millennium of Christianity. Infallible truths produced by natural reason has led to false pronouncements and a failed program which contributed to Western skepticism and the death of God (cultural); ideas have consequences and the failure of this idea has had momentous consequences in western philosophy and culture. So for us as for you it is not simply a matter of accepting or rejecting elements of scholasticism, but accepting or rejecting elements of scholasticism AS CONSTITUTING A DOGMA OF THE CHRISTIAN FAITH. Many papal/ teachings assert X, Y, Z is a matter of moral law perceptible to natural reason apart from faith. This is originates from Aristotle, is taken up by Aquinas, and baptized by the Magisterium, but it is not found at all in the teaching of the Undivided Church before the Great Schism and before the Hegelian theory of dogmatic development we find at the forefront of Roman Catholicism from Newman to Vatican II and beyond (cf. Novus Ordo liturgical revolutions etc.).
Turning from questions of dogma to matters of intellect and/or conscience one hardly knows where to begin.
One reason for rejecting scholastisism is it is most often regarded on its own chief grounds, reason, as failed and unconvincing; outside of traditionalist RC circles scholasticism and Classical Foundationalism are effectively dead in philosophy:
"Foundationalism has been the reigning theory of theories in the West since the high Middle Ages. It can be traced back as far as Aristotle... Aquinas offers one classic version of foundationalism. There is, he said, a body of propositions which can become self-evident to us in our present earthly state. Properly conducted scientific inquiry consists in arriving at other propositions by way of reliable inference from these (demonstration). A few of these (for example, that God exists) can be inferred from propositions knowable to the natural light of reason.
...within the community of those working in philosophy of knowledge and philosophy of science foundationalism has suffered a series of deadly blows in the last 25 years. To many of those acquainted with the history of this development it now looks all but dead. So it looks to me. Of course, it is always possible that by a feat of prodigious imagination foundationalism can be revitalized. I consider that highly improbable..." (Nicholas Wolterstorff, Reason Within the Bounds of Religion, pp. 26-27).
Contrast the Vatican Insider's claim: "In a pluralistic society, the Catholic Church is convinced that it is duty bound 'to intervene in favor of the values that are valid for man as such, independently of the various cultures' - values the Church knows 'through its faith' but at which all men can arrive through reason alone, regardless of faith." http://vaticaninsider.lastampa.it/en/the-vatican/detail/articolo/benedetto-xvi-benedict-xvi-benedicto-xvi-9725/
With a majority of my contemporaries who have studied ethical philosophy at the graduate level I for one thank God for Orthodoxy, lest throwing the baby out with the bathwater I would be faced with the false dichotomy of embracing the Apostolic Faith AND an essentially dead, or at least highly dubious according to a decided majority of professional philosophers of our day as if they must be imbibed together.
Catholic theologian Charles Curran claims "the concept of natural law as a deductive methodology based on eternal and immutable essences and resulting in specific absolute norms is no longer acceptable to the majority of Catholic moral theologians writing today" (Curran, Charles, "Catholic Moral Theology Today" in New Perpectives in Moral Theology, ed., Charles Curran (Notre Dame: UNDP, 1982), p. 6). A standard circular objection to the fallout among Roman Catholic philosophers is that this is "dissent"; here however we should note that Curran is simply stating the fact of its general rejection, which is a matter of simple statistics.
Since the question was about an Orthodox reaction
"Medieval scholasticism argued for the existence of God in the chain of causation: God as First Cause or Prime Mover. This is quite problematic since God does not belong to the category of facts. He is not a fact among facts and cannot be considered in such a manner. We may follow a chain of causation and arrive at what we cannot know. For some, this constitutes proof. For others it begs the question." http://glory2godforallthings.com/2013/01/22/the-beauty-of-truth-and-the-existence-of-god/
Cf. also several specific objections cited in a recent article by Orthodox writer David Bentley Hart http://www.firstthings.com/article/2013/05/nature-loves-to-hide
and also http://ethikapolitika.org/2013/03/14/hart-has-reasons-that-reason-cannot-know/
These article form a part of a recent debate stimulated by Hart which will be instructive to readers on both sides of these questions.
Natural law theories as an example are purportedly a matter of logic; historiography demonstrates otherwise. Natural law theories and their content are culturally conditioned. "...every attempt to spell out the intellectual content of natural law can be shown to be historically and culturally conditioned. While all people seem to have a moral sense, when they begin articulating what this means, their own cultural and religious background proves to be determinative in their judgments. We need to take seriously this telling criticism of Erik von Kuehnelt-Leddihn: 'Is there such a thing as a natural law in the sense that we all 'naturally' reject murder, lies, deceit, wanton cruelty, adulterary, theft, or contempt of parents? As a world traveler and student of ethnology I deny this in the face of certain Christian theological tradition " Erik von Kuehnelt-Leddihn, "Jews, Christians, and Gentiles," National Review 35, no. 20 (Oct. 14, 1983), p. 1282).
No less important are the numerous religious objections to scholastic philosophy as it has been embraced dogmatically by Roman Catholicism.
Scholasticism has been rejected by Protestants like Barth and Bonhoeffer (as by the Reformers before them) as being in stark contradiction to the Christian faith. In the Genesis narrative knowledge of the good -far from being part of creation/nature, appears as a part of the fall. The tree in the garden man was forbidden to partake of was not simply the tree of the knowledge of evil, but the tree of the knowledge of good as well. Knowledge of the good was in scripture part of the fall, not part of creation or nature or humankind It was on this view "the devil's first lie" that we can have knowledge of the good" apart from revelation or union/communion; Aquinas accordingly might seem to have simply taken up the argument of the devil who on this view must have rejoiced beyond measure when it was pronounced a dogma by a modern pope? In any event Genesis does not especially remind us of Aristotle or the Stoic philosophers concerning natural knowledge of the good.
Natural law proponents appeal to St. Paul concerning the law written on our hearts. Although scholarship and exegesis hardly decide this matter it is well to remind ourselves of the peculiar fact that those who see the Pauline texts teaching "natural law" as the Roman Catholics understand it are *almost entirely* Roman Catholics. Pascal famously argued that the heart has its reasons which reason knows nothing of. These laws written upon the heart Paul spoke of may be congruent to reason (e.g. one might describe them), but it is not self-evident they are *arrived at by reason* in the sense of a "foundation" or as a process of discursive or syllogistic reasoning in the sense described by Thomas Aquinas et al (cf. the Natural Law trajectory in philosophy from the Stoics, Aristotle, Aquinas, Roman Catholicism, Neo-Thomists, etc.).
No doubt our Thomist friends have some counter-reactions and objections to some of this, which is fine; if these brief remarks serve to make readers aware that rather than there being some marginal Orthodox rejection of a Grand Idea that is secure in other quarters we should rather understand an increasingly marginal Roman Catholic dogma rejected not only by a few Orthodox Christians, but also most major philosophers who are not Roman Catholic as well as being totally absent from the entire first Christian millennium.
Supposedly, according to traditionalist Roman Catholic replies, we should not judge this issue according to how many philosophers affirming Natural Law ("objectively true moral conclusions ...derived from premises that in no way presuppose any purported divine revelation, any body of scriptural writings, or any particular religious tradition" -Hart, op cit) can be found who are not traditionalist Roman Catholics. Nevertheless the issue seems glaring:
When a thing "perceptible to reason apart from faith" is also rejected as obvious from natural reason by not just a majority of contemporary philosophers, not just Orthodox Christians "in the 20th century," not just most Protestant writers from the Reformation to the present, not just Jewish writers, not just a majority of contemporary Roman Catholic philosophers as affirmed by Curran, but a majority of practicing Roman Catholics themselves (recent poll data) what might otherwise be dismissed as marginal or moderate suspicion appears to be approaching critical mass.
Scholasticism is merely this: the use of reason to seek whether or not certain aspects of faith may be strengthened through reason.
Strengthening of faith through reason is one thing; the claim that reason may be considered as a foundation upon which "the whole science of natural and divine things is based" is quite another:
"The capital theses in the philosophy of St. Thomas are not to be placed in the category of opinions capable of being debated one way or another, but are to be considered as the foundations upon which the whole science of natural and divine things is based; if such principles are once removed or in any way impaired, it must necessarily follow that students of the sacred sciences will ultimately fail to perceive so much as the meaning of the words in which the dogmas of divine revelation are proposed by the magistracy of the Church. -Pope Pius X, Doctoris Angelici (29th June 1914).
I cannot underscore enough how right ialmisry has it:
For St. John (and the rest of the Fathers) philosophy provides a tool to organize thoughts and systematize data. For the Scholastics philosophy can provide new data: hence the elaboration of natural law. For St. John the Fathers then philosophy is always theology's handmaiden, for the Scholastics theology's younger sister, if not her twin. St. John, and the Fathers, feel no need to use philosophy to stretch knowledge beyond revelation. The Scholastics attempt to use it to not only know the unknown, but the unknowable.