We have two or three threads going on the topic of the primacy of Peter and Rome, but the main one (forums.catholic.com/showthread.php?t=19413 ) has sort of been left on the wayside and has become too large anyway.
This time I want to try to stay focused on the Fathers, and try not to be diverge from them too much. Ultimately, while Scripture exegesis is very important, the true One, Holy, Catholic, and Apostolic Church must be verified by the Fathers. If it is the Eastern Orthodox Church, we should see the majority of the Fathers granting Rome, at best, a primacy of honour, and never insisting that the primacy is of divine institution. If the Catholic Church is right, the Fathers should, as a whole, suggest that the primacy of Rome is more than just honour. (Though it should not be necessary to find a full-blown modern Roman Catholic understanding of the papacy in the Fathers, as we do advocate the development of doctrine).
I’m going to start by providing a few quotes from web.globalserve.net/~bumblebee/ecclesia/patriarchs.htm. I highly recommend the rest be read as well (just follow the link). All of the following quotes, and the rest on the page linked above are Eastern Christians from the first millenium. These are only a sample of the ones found on the above page.
I therefore beseech your holiness to persuade the most holy and blessed bishop (Pope Leo) to use his Apostolic power, and to order me to hasten to your Council. For that most holy throne (Rome) has the sovereignty over the churches throughout the universe on many grounds. (Theodoret, Tom. iv. Epist. cxvi. Renato, p. 1197).
(Theodoret, Bishop of Cyrus in Syria, 5th century)
(Peter), the foundation of the Church, the Coryphaeus of the choir of the Apostles, the vehement lover of Christ …he who ran throughout the whole world, who fished the whole world; this holy Coryphaeus of the blessed choir; the ardent disciple, who was entrusted with the keys of heaven, who received the spiritual revelation. Peter, the mouth of all Apostles, the head of that company, the ruler of the whole world. (De Eleemos, iii. 4; Hom. de decem mille tal. 3)
And why, then, passing by the others, does He converse with Peter on these things? (John 21:15). He was the chosen one of the Apostles, and the mouth of the disciples, and the leader of the choir. On this account, Paul also went up on a time to see him rather than the others (Galatians 1:18). And withal, to show him that he must thenceforward have confidence, as the denial was done away with, He puts into his hands the presidency over the brethren. And He brings not forward the denial, nor reproches him with what had past, but says, 'If you love me, preside over the brethren, …and the third time He gives him the same injunction, showing what a price He sets the presidency over His own sheep. And if one should say, ‘How then did James receive the throne of Jerusalem?,’ this I would answer that He appointed this man (Peter) teacher, not of that throne, but of the whole world. (Chrysostom, In Joan. Hom. 1xxxviii. n. 1, tom. viii)
(St. John Chrysostom, Patriarch of Constantinople, 4th century)
Peter, the coryphaeus of the disciples, and the one set over (or chief of) the Apostles. Art not thou he that didst say, ‘Thou art the Christ, the Son of the living God’? Thou Bar-Jonas (son of the dove) hast thou seen so many miracles, and art thou still but Simon (a hearer)? He appointed thee the key-bearer of Heaven, and has though not yet layed aside thy fisherman’s clothing? (Proclus, Or. viii In Dom. Transfig. t. ix. Galland)
(St. Proclus, Patriarch of Constantinople, 5th century)