"Raumschach" (German for "Space Chess") is the most successful version of 3D chess. It's played in a 5x5x5 space. The rules are analogous to regular chess, except there's no en passant, there's an extra piece called the unicorn, and the pawn moves need to be learned in order to play.
On this website, you can take a tutorial to learn the moves and play against a computer opponent. In the near future, rated and unrated correspondence play will be implemented. If that interests you, bookmark the page and check back regularly!
Below, you can see the starting position on the left, and a middle-game position on the right. Enjoy playing!