Poker
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There are several sites that use play money. Fulltilt, Ultimatebet, and Poker Stars just to name a few. All of these sites have real money tables, but they also cater to the non money players as well. Personally I like Fulltilt for the graphical interface. Hope this helps.
Thanks Tim! I just downloaded Full Tilt Poker and starting to play (somewhat). The program is only for Windows and Mac (I hope there'll be a Linux in the future). 
There is an upsurge of interest in the game of poker, maybe particularly the holdem variety. I'm posting this question here because I think chess players are also poker players. Do you know of a poker website with similar functionality as Chess.com? Maybe like playing online without using real money (of course with an option to use real money). I'm not into gambling; just for the fun of the game. I've gone to many sites but you have to bet with real money.