You always hear about how the Muslim empires preserved Greek and Roman learning and art, but what about the Byzantines? Wouldn't they have the most direct link to Greece and Rome during the Middle Ages?
Your intuition is correct.
Byzantium had a number of philosophers who wrote commentaries on Aristotle and preserved the works of the Greeks. The reason we generally don't think of them when discussing the history of philosophy is because of the development of hesychasm in the East. If you are unfamiliar with hesychasm it is a mystical form of prayer. This emphasis on mysticism created even greater contrast with the West who became focused more on philosophy, particularly Scholasticism.
In fact, a large part of the Renaissance in the Western Church can be attributed to Eastern intellectuals (such as Barlaam) and various Greek scribes and printers moving west in reaction to this new emphasis on mysticism (as well as the decline of the Byzantium empire). They brought with them knowledge of the Greek language as well as the writings of the Greeks themselves.