great idea!
Chess.com Xbox 360/Xbox One
You can login to Chess.com and play games via Xbox 360 Internet Explorer, but Blitz is impractical without a mouse.
Also, just found this on Amazon.com.
http://www.amazon.com/Battle-vs-Chess-Xbox-360/dp/B0035FYXS2
Chess.com would have to hire developpers for this kind of project and then maintain it. There is a cost involved and if they do it for Xbox, they might to do it fro PS3-4 as well. Is there enough demande for this kind of make to make such project sustainable and would people be ready to pay for it?
I would pay $5 to $10 for it. Prefer playing PC ports on Xbox 360 console (i.e. Xcom, Civ Rev, Portal 2) and kicking back on on couch instead of sitting at chair in front of PC.

Rejoice! Even though Internet Explorer for XBOX360 does not support Flash, and makes it impossible to play on chess.com or lichess.org, there is one site which apparently does support the browser. Open your XBOX Explorer and try [removed external link - JG]. You should be able to play against humans as "guest" without even registering. Let me know how it went.
Anyone here wish Chess.com was available as app for Xbox 360/ One Platforms? Also, I don't see any great chess software for Xbox 360/One. However, Chessmaster 10th edition was produced for Xbox original.