There's a lot of research done on this topic, and a quick google-search brings up a couple of interesting articles.
For me personally, I'm not committed either way right now, though I lean heavily towards the belief that Christ did indeed speak Greek. This is all, once again, from rereading Mark. It's not conclusive, but here's why I'm leaning.
It comes down to the explanatory scope of the two hypothesis regarding the two passages in Mark where Christ is quoted as speaking in Aramaic. Now, my pastor says that there are places where it is recorded that Jesus spoke "in Aramaic" (namely, that there are places where it makes a special case pointing out that here he speaks in Aramaic), but I can't seem to find one. But here are the two places where Jesus does speak in Aramaic -
Mark 5:41 (Little girl, arise) and Mark 15:34 (My God, why have you forsaken me?).
The question for me is "which hypothesis fits the two texts better?"
If we assume that Jesus normally spoke in Greek, we have an easy explanation of Mark 5:41 (maybe it was the language the girl understood), but we have a little more difficulty with Mark 15:34 (It's not like that's the only language God understood). My best hypothesis was that Christ was (once again), fulfilling the "let he who has ears, hear" idea one last time, where the secret of the Kingdom would be hidden from some and given to others. Most of the crowd did not understand, but it seemed that the one centurion did. It's not clearcut, but at least it is possible.
With the other hypothesis however, if Jesus normally spoke in Aramaic, we can explain the Mark 15 passage by saying that usually Mark translated, but here he did not because he wanted to explain the crowd's confusion regarding Eloi and Elijah. But I cannot even begin to conceive of why Mark quotes Jesus in Aramaic instead of just Greek like he has everywhere else if Jesus normally spoke Aramaic.
So because of that difficulty, I'm leaning towards Greek, but maybe reading other Gospels will change my mind. I'd love to know what others think, if they're still reading this blog. =P