Fortunately, they're coming, probably as soon as Apple can approve them. Touch Arcade forum member Project-79 has been keeping track of which games have said that they'll be Game Center-enabled. As soon as updates roll out for games like Field Runners, Flight Control, Super Mega Worm and Angry Birds, you'll be able to jump in and earn achievements with them.
Keep in mind that some of the best games on Game Center are still to come: Gameloft's Dungeon Hunter 2 was leaked via a Game Center screen, and of course we're all waiting to see what Epic Games' Project Sword looks like, ever since it was shown off with full GC functionality at last week's Apple event. There are games coming for Game Center, and they should be here soon.
Update: Apparently Ms. PAC-MAN is the first game to get updated with Game Center support. You might want to wait, though -- we've heard it's not a very good version.