The Greenway going downtown along the East Side, starts a good bit lower than the Queensboro Bridge. I believe it starts after the U.N. which means it would start around 41st or so.

Actually just checked the map provided above. The greenway along the East Sides stops at the QB Bridge and does not pick up again until 37th Street. They have bike paths (not too useful ones though) on 1st & 2nd avenues though.