The question of Filioque is very delicate and needs some study here, so that you might understand the question in its depths.
First of all, the Latin and Greek traditions of the Church Fathers were originally in full agreement and mutually complementary. The Greek Fathers, being used to read the Scriptures in the original language, read the sentence from Jesus "But when the Paraclete cometh, whom I will send you from the Father, the Spirit of truth, who proceedeth from the Father, he shall give testimony of me. " (John 15:26) they noticed the original verb for proceedth, i.e. "Ὅταν δὲ ἔλθῃ ὁ παρὰκλητος, ὃν ἐγὼ πέμψω ὑμῖν παρά τοῦ πατρός, τὸ πνεῦμα τῆς ἀληθείας, ὃ παρά τοῦ πατρὸς ἐκπορεύεται, ἐκεῖνος μαρτυρήσει περὶ ἐμοῦ·". This verb means something like "to spring out from", "to be originated". When the original Creed approved of at Nicaea was to be expanded at the Second Ecumenical Council, the Church Fathers incorporated this sentence in the text regarding the Holy Spirit, in order to affirm - from a certain Biblical authority - what was known of the Paraclete: that he was God originated from God the Father. The same expression was often completed by other Fathers using the addition "through the Son".
Meanwhile, there was a secondary concept of procession forming in the Latin and Alexandrian traditions. This is not the same as the procession of origin which is established as a dogma at Constantinople I. St. Cyril of Alexandria wrote that the Holy Spirit "proceeds from the Father and the Son" using the verb "proienai". This form means something like "to be sent forth", and was similar to the original Latin form "procedere". But here's the main problem. The Latin Fathers were correctly using the verb "procedere" in the cyrilline sense, and we have proofs of this from st. Ambrose of Milan. Yet, this verb was also used incorrectly to translate the verb ἐκπορεύεται in the Creed, creating a confusion, a victim of which was st. Augustine of Hippo, who felt that - according to the Latin tradition - the Creed developed in Constantinople was right but yet incomplete. Augustine affirmed originally a position similar to the Greek "through the Son" when he wrote that the Spirit "proceedth from the Father principaliter (by principle) and from the Son secondarily". Yet, he confused the proper individuality of the Paraclete with the common "theotes" (=divinity) of the Father and the Son, and more precisely with God's energy called Love. This was an unprecedented error which was to damage RC theology at a later time.
Anyway, the synod of Constantinople was granted an ecumenical authority at Ephesus, and this council reaffirmed the ecumenicity of the Creed "of Nicaea" (in its expanded form, of course) and the impossibility to modify it outside of an ecumenical context.
Meanwhile in Spain was convened the anti-Arian synod of Toledo. Spain was being delivered from the Arian cancer and was alligning to mainstream Catholicity, so they condemned the submission of the Son to the Father affirming, in a legitimate Latin formula, the "procession" of the Spirit from Father and Son as it was meant by st. Ambrose too. Nevertheless, this version of the Creed spread to Gaul and was to be later considered by Charlemagne to be the only full Creed of the Church in the 9th century.
The Church of Rome, while approving both the Greek and Latin creeds, had no tradition of singing the Creed at Mass. At the 7th Ecumenical council, held in Nicaea, the Church Fathers reaffirmed the Creed in its original Greek form, and added as an explanation that by "proceedth from the Father" the expression "through the Son" was implied. On the base of preserving Church unity, pope st. Leo III - a valid successor of st. Peter - opposed to the claims of Charlemagne who wanted the Creed with Filioque to be introduced in Rome, so he made two tablets be hung in Rome, exposing the original Greek Creed and its Filioque-less translation in Latin. The question seemed to be set once and for all, but it wasn't so. Rome was later pressed by the Holy Roman Empire (presumably for political reasons) to affirm the Filioque clause and add it to the Greek Creed. The leader of the Greek anti-filioquist party was st. Photius, patriarch of Constantinople, who was deposed by the Pope and replaced by a certain Ignatius appointed by the Roman Pontiff. In 870 a council held by only 12 bishops and some papal legates reaffirmed the papal decisions, and the two branches of the church underwent a temporary breach. This synod is known to RCs as the "8th Ecumenical council". Only ten years later, a counter-council was held, also in Constantinople, with some 380 bishops and papal legates present. This one condemned the dispositions of the previous 870 AD synod, and imposed that no addition, omission or alteration of the Creed of Nicaea (but the text given is that of Constantinople I!) could be incorporated. The same synod stopped the claims of the Pope for super-jurisdiction and re-enthroned Photius on his patriarchal see. The Horos (=decree) of the Council was translated in Latin and read by pope John VIII, who was benevolent towards the Easteners (he showed great understanding for the cause of Cyril and Methodius, apostles of the Slavs), signed for the reinstallment of Photius, thus giving ecumenical authority to the Synod.
Nevertheless, in 1012 AD pope Benedict VIII was enthroned, and he was asked at the crowing of the new Western Emperor to sing the Creed with Filioque, and he did. In 1054 AD, during a sedevacante, the Latin dogma of Filioque was even imposed by cardinal Humbert of Silva Candida to patriarch Michael Cerularius, who was excommunicated, and a counter-excommunication followed. Despite all attempts of dialogue from the Eastern part, in 1098 AD (the year preceeding the first crusade!) pope Urban II convened a pseudo-council at Bari, Italy, and finally sanctioned the heresy of Filioque and papal supremacy, and all dialogue ended once and for all.
There are at least four reasons why the Orthodox reject the Filioque clause.
1)The Filioque is uncanonical. The original Creed was affirmed as unchangeable at Constantinople I, Ephesus, and Constantinople IV, and the only allowed catechetical explanation/addition allowed is the formula "through the Son" sanctioned at Nicaea II. Changing the Creed is against the common Canon Law of the Holy Orthodox Catholic Church.
2)The Filioque is not Biblical and Traditional. There is no single verse in Scripture or in the writings of the Church Fathers where the Greek verb "ἐκπορεύεται" was combined with the Filioque, while the expression "from the Father" is clearly written in John's Gospel.
3)The Filioque is heretical. To proceed, in its Greek original, means "to be originated". The purpose of the Creed is to define the "only" arché (principle) of the Paraclete's personality and divinity, what the Church Fathers called "Monarchy of the Father" within the Trinity. Also, confusing the Spirit with the common love (an energy) or nature (divinity, or theotes) of the Father and the Son is also a serious theological error.
4)The Filioque was used as a political and religious instrument from the West to affirm the superiority of the Pope over the other bishops and a supposed divine authority to the Holy Roman Emperor over the legitimate Roman Emperor of Constantinople. This also menaced the concilary nature of the Church in defining dogmas.
Hope this helps,
in Christ, Alex