Do you have "mass intentions (offering Masses/Divine liturgies for a specific person)"?
I think that the closest thing that you will find in the Orthodox Church is the uses of prosphora during the Proskomidia. In the Russian tradition and I'm sure others the use of small loafs of bread called prosphora are blessed and a small portion of the loaf is taken out in commemoration for the living and the dead and used as part of the bread that reserved for the Eucharistic mystery. In my church every family usually has a commemoration list for their living and reposed
family members and friends which gets sent back into the alter with a prosphora loaf for a blessing and for a particle to be cut out for communion.
The most effective form of prayer for others is doubtless the eucharistic prayer. At every Eucharist celebrated in our churches, we have the possibility to bring under the saving power of the Eucharistic Sacrifice all those we care for, present and absent, living and dead. Every Eucharist, every Holy Liturgy is a repetition of the Sacrifice of Golgotha. Christ's blood is shed and sacrificed again for the sins of the world and of mankind -- and for their salvation The bread brought to the altars of our churches becomes Christ's Body. Partaking of Christ's Blood and Body, we are united with Christ; we receive the food of eternal life; we mature for the Kingdom of Heaven. The Blood of Christ washes away our sins, cleanses us of the dirt of the world -- and not only us who are present at the liturgy but all those whom we wish to include into the saving action of this Sacrament. That is why, at the end of every liturgy, as the priest dips the particles from the paten into the chalice, he prays: "WASH AWAY, O LORD, THE SINS OF ALL WHO ARE HERE COMMEMORATED by Thy precious Blood, through the prayers of Thy Saints." This is where the power is born that saves from under an onrushing car, from a bullet aimed at us, from the evil lying in wait for us. The particles of PROSFORA (altar bread) remaining on the paten after the communion of the clergy and laity represent, besides the Mother of God and all the Saints, all those (living and dead) for whom the priest and the lay people present wish to pray for at the eucharistic prayer. These particles are taken by the priest from the prosforas sent to the altar with the remembrance lists of parishioners. Every name is mentioned separately, and a particle representing the soul of the living or deceased person is taken out of the corresponding prosfora. These particles remain in the paten below the sacrificial bread, which becomes the Body of Christ during the liturgy. At the end, the priest dips all these particles into the chalice containing the Holy Blood, saying the aforementioned prayer. Besides these particles, those taken out in honor and commemoration of the Mother of God and All Saints are on the paten. Thus, the soul of each man whose name is mentioned at the liturgy is mystically incorporated into the realm of the salvation of the Church. As St. Paul said in the epistle of the Ephesians: "Now therefore ye are no more strangers and foreigners, bur fellow citizens with the Saints, and of the household of God" (Ephes. 2:19). In the Ancient Church, the Christians brought the bread and wine for the Eucharistic; that is why these are called "gifts." We give them to God as a part of our work and efforts, a fruit of our harvest. The Lord returns them to us as His "Holy Gifts," given us for our life and salvation through His grace and mercy. In our days of "civilized" city life, man has unintentionally lost his ties with the earth, and his eucharistic gift is the bread that he buys at the candle counter. Yet this bread has preserved the whole meaning of the eucharistic gift of sacrifice and is thus still called "prosfora," which is the Greek word for "offering." The eucharistic bread is distinguished by several symbolic features: to separate it from the Holy bread of the Old Testament, it is made with yeast ("Therefore, let us keep the feast, not with the old leaven" - 1 Cor... 5:
, its round shape is a reminder of the eternity of the life and of Church; its two parts testify to the Divine and human natures of Christ. The top of the prosfora bears a seal depicting a cross, the initials of Christ, and the Greek word NIKA, meaning victor.
Every Orthodox family or single person is supposed to have a remembrance list - a sheet or booklet listing the names of the members of the family, relatives and friends, dead and alive, for whom the family or person wishes to pray. In composing a remembrance list, only the Christian names should be used. For example, Mary Smith (or just Mary) must be used instead of Mrs. John Smith. This list is then given to an attendant at the candle counter and then is sent to the altar with a corresponding prosfora. The first part of the Divine Liturgy, called the PROSKOMIDIA (Greek for "oblation"), is dedicated to the preparation of all that is needed for the liturgy and to the eucharistic prayer for the living and the dead. At a certain moment during the Proskomidia, the priest prayerfully reads all the names on the lists, taking out and depositing small particles of prosfora onto the paten for each name. At the end of the Proskomidia, he reads the prayer of oblation, asking God to remember all "those who offer it and those for who it is offered": those who have brought the lists an eucharistic bread and those for whom they were brought. This prayer for these souls having need of God's help in this life and of Divine mercy in the eternal life goes on throughout the liturgy and reaches its highest point when the prosfora particles are dipped into the chalice with the Divine Blood, washing away the sins of the living and the dead.
The eucharistic prayer calls for early arrival at church, so that the remembrance lists and prosforas may be brought to the altar during the Proskomidia, or at the very beginning of the Liturgy of the Catechumens. However, it is possible to leave the list and the donation for the prosfora the night before the liturgy.
After the liturgy, all those who have given their lists and prosforas get them back at the candle counter. These prosforas are not just bread, but bread which has been sanctified through being a part of the sacrament of the Eucharist. A prosfora is either consumed immediately after the liturgy, or taken home, broken and distributed to the family at the first meal after the liturgy. There is no doubt that partaking of the prosfora with faith and prayer contributes to the mental and physical health of those who partake. Every Orthodox family (or every Orthodox Christian) should have a remembrance list. Names on it will vary and change places according to occurrence of births and deaths in the family. The more often such a list is presented for eucharistic prayer, the more Divine mercy and protection will encircle the family and all those remembered.
The Right Reverend George M. Benigsen
The Orthodox Church, June 1979, p. 6.