I've looked in Game Center on Mountain Lion and in the Chess game, but I can't seem to see a way to play Chess over Game Center. Does Chess for Mountain Lion support turn-based play? If it does, how can I enable it?
First of all, make sure you have added your systems knows about your Apple ID (go to System Preferences -> Users & Groups -> Your User -> Apple ID).
Then launch Chess; obviously you'll need to have an internet connection.
After a short wait (a second or two in my case, might depend on your internet connection), a badge like the following should appear in the Chess window:
After it appears, you should simply be able to go to Game -> New... and select GameCenter Match in Players.