What would happen if you could dig a hole straight through the center of the earth to the other side?

Random question, but I've wondered this from time to time since I was young. I always figured that, even if you were to actually get through the core (don't be that guy to call the molten core detail), you'd eventually be stuck in a falling state as gravity pulls you toward the center, the momentum would propel you "up" the other direction, and you would fall the other way, maybe ad infinitum.

Anyone more educated in this kind of thing have an answer?