The link is direct between the Eastern Orthodox Church and the New Testament Church in Jerusalem. The current Patriarch Theophilus III can trace himself via apostolic succession directly to St. James the Just in the 1st century AD.
I did a powerpoint for my church on the Holy Fire which comes down every Holy Saturday in the Holy Sepulcher and included some of the history. I will repost some of it below but unfortunately without the pictures (which actually is the best part of the presentation

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Severe persecution in the first 100 years of the Church. Communities scattered and became refugees.
The Decapolis (“Ten Cities”) were a large province near Galilee which where Hellenistic.
In one of the these cities, the city of Pella, the Church of Jerusalem found refuge and it became the new seat for the Church. (that is, the one and only Bishop of Jerusalem, in exile with the Christians who fled the city, chose this city to settle in to protect the flock)
Here it began to purge itself from certain Judean elements which ran counter to the teachings and gospel of Christ.
AD 135, the Roman emperor Aleius Hadrain sent his general Severus to Jerusalem to squelch the second great revolt of the Jews against the Romans.
Jerusalem was decimated, and a new city was formed nearby named Aelia Capitolina.
The Tomb of Christ was filled with earth and a temple of Aphrodite was built on top of it.
Quadratus and Aristides were brilliant Roman philosophers (considered the Einsteins of their day) who became Christian apologists (and eventual Bishops) and convinced the Roman Emperor Hadrius to protect the Christians on account of the virtuous lives they lived and the unfair persecutions they were undergoing. The Emperor passed a decree which forbade Christians to be put to death unless accused in due form and convicted of offense against the laws and also punished the accusers if their charges were found to be false. This offered protection to the Christians. (see
link)
With the rebirth of the holy city and a reprieve in the heavy persecution, the Church which had fled Jerusalem had now returned, now more Hellenized in character but retaining the same apostolic doctrines and faith and eucharistic sacramental communion with all the other churches across all of Christendom.
The election of the next Bishop in line in the seat of St. James the Just was Bishop Marcus in the year 135 who was of Greek descent (Jewish born were not allowed to enter the city of Aelia Capitolina/Jerusalem by Roman Emperor decree).
From then on, the Patriarchate of Jerusalem has never left and is the guardian of the most holy Christian site in the world, namely the Church of the Resurrection which houses the Holy Sepulcher (the tomb where Christ resurrected)
The Patriarchate of Jerusalem was granted autocephaly in 451 by the Council of Chalcedon and in 531 became one of the initial five patriarchates (The Pentarchy of the One, Holy, Catholic and Apostolic Church).
The Eastern Orthodox Church is the same Church of Jerusalem we read about in the New Testament, historically, spiritually ,sacramentally, and doctrinally. It is the One, Holy, Catholic and Apostolic Church which has carried the faith of the apostles.
I read this on the internet and it is a nice short way to sum it up: The Orthodox Church is evangelical, but not Protestant. It is orthodox, but not Jewish. It is catholic, but not Roman. It isn't non-denominational - it is
pre-denominational.
It has believed, taught, preserved, defended and died for the Faith of the Apostles since the Day of Pentecost 2000 years ago.
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