Curtis Granderson over the last three seasons posted a weak .223/.280/.334 slash line against left handed pitching. The reason may be is that he is hitting against the way pitchers work him.
Left-handed pitcher work Granderson away in two dimensions. They locate the ball on the outside of the plate, plus the movement he most often sees has the pitch moving away as well. If you look at his hit chart, however, it's clear Granderson tries to pull pitches. In general, pulling the ball works well. A batter generates more power, and probably hits the ball harder when he pulls. It's not working for Curtis, however. Often times he just grounds out to the second baseman. Although it may be difficult at this point in his career, learning to smack the ball the other way against lefties might not only raise his averages, it may make the southpaws more honest so they pitch inside a little more often where Granderson can do some real damage.