I'm an Orthodox Christian. The Orthodox believe (and this goes back to very early Christian times) that Christ is the fulfillment of the OT. He even says this himself. The OT law was a "shadow" of things to come. There's lot of this in the Epistle to the Hebrews.
The OT is a very important part of Tradition in the Orthodox Church. In it you find prefigured many of the events and many major people in the NT, not the least Christ and the Theotokos (Virgin Mary - Theotokos is Greek for "Mother of God).
For example, the Suffering Servant in Isaiah in points to Christ. Jonah is also a type of Christ, because he was in the belly of the large fish for three days, prefiguring Christ's being in the tomb for three days. Christ even uses this one Himself - telling the Jews against Him that they will receive no other sign than that of Jonah - Christ rising from the tomb after three days.
Joseph the Patriarch in the OT is also held to be a type of Christ.
Christ Himself taught how the OT should be interpreted in light of Himself on the post-Resurrection appearance on the road to Emmaus (Luke 24:25-27) and to the Apostles (Luke 24:44-48). He also referred to Himself as the fulfillment of the Law.
Melchizedek is a type of Christ as well.
Psalm 1, "Blessed is the Man," speaks of the perfect man that follows God's ways - that is Christ.
Psalm 45 foreshadows the Theotokos, as does the Burning Bush in Genesis, Ezekiel 44:1-4, the dewy fleece in Judges 6:36-38, and of course, Isaiah 7:14.
You simply can't have the NT without the OT. You need to have read and understood the OT to truly get the NT. Otherwise, a lot of stuff doesn't make much sense (such as the Epistle to the Hebrews, which talks much about worship in the Jerusalem Temple) or you'll miss the meaning of. Psalm 22, recited on the Cross by Christ, "My God, my God, why hast Thou forsaken me?," foreshadows the Crucifixion.
God's resting on the seventh day - the Sabbath - after he had completed the work of creation, foreshadows Christ's resting in the tomb on the Sabbath, after He had completed His work on earth.
The OT Passover foreshadows the Resurrection of Christ. When the Hebrew Bible was translated into Greek about 250 years before the birth of Christ (this translation is known as the Septuagint), the Hebrew Pesach, the word used for Passover, became the Greek Pascha - the word the Orthodox Church still uses for the Feast of Feasts, Christ's Resurrection. Also see 1 Corinthians 5:7-8, "Christ, our Passover, was sacrificed for us. Therefore let us keep the feast." Christ is the unblemished Paschal Lamb that was sacrificed for us, with none of His bones broken. See Exodus 12:10 and John 19:33, 36.
These interpretations of the OT through the lens of Christ are nothing new. They are very ancient. This is the traditional way the OT has been interpreted by the Church Fathers.
BTW, the quotes from the OT used in the NT are from the Greek OT (Septuagint). The Septuagint text varies in many places from the Hebrew text, particularly in the Psalms and the Prophets. I've done comparisons myself putting the new (2008) English translation from the Septuagint up against an English translation from the Hebrew (such as the RSV), and there is a lot in the Septuagint that is simply not in the Hebrew.
In the Nicene Creed (the largest portion of which came out of the First Ecumenical Council in 325 AD - the last section from the Holy Spirit on, were added at the Second Ecumenical Council), we say Christ "and on the third day He rose again according to the Scriptures" - those Scriptures were the OT. In various places in the NT, where St. Paul speaks of the Scriptures, he's also referring to the OT.
Early Christians who discounted the OT were known as Marcionites:
http://en.wikipedia.org/wiki/Marcionites
