Personally, I would consider an lcd for gaming use only if the room is super bright with no possibility of curtains and that's the only time you can play. I completely agree with Rodolfo's comments and would add that you should also be aware of lcd (tv or monitor) pixel response times (how fast the pixels turn off and on) and refresh rates (how many frames per second it displays).
If you decide on an lcd, most gamers want a <= 2ms response rate. Further, a low frame rate tv/monitor could throttle the graphics hardware. With game applications, higher frame rates and higher frame resolution both demand more machine graphics, compute and memory resources (regardless of your tv/monitor display) and will affect game speed and picture quality. Currently the best lcd tv frame rates max out at a true 240 frames per second (fps), (with lots of misleading manufacturer hype).
These issues are mostly moot, however, for the Xbox 360:
1) most games on the Xbox 360 system render frames at 30 fps and the Xbox then scales the frames to the display resolution;
2) 720p is the only feasible resolution to run an Xbox 360 at, as most games become unacceptably slow at 1080p resolution.
If you do get a 1080p monitor/tv, let it do the scaling, not the Xbox. A list of many Xbox 360 games' native resolutions can be found at: http://forum.beyond3d.com/showthread.php?t=46241
Your decision will depend largely on your budget. If you can afford it, plan for the future and get a 1080p plasma tv (generally cheaper than a true monitor), put up with any lighting situation you cannot correct, and save up for a game machine that can tax the plasma screen; otherwise, I would recommend looking for either a Samsung or Panasonic 720p plasma tv which are a steal compared to the same sized 1080p sets. I say Samsung (slightly better in brighter lighting) and Panasonic (slightly better in low lighting) because I have personal experience with them and would recommend them. Rodolfo has experience with many more manufacturers and may well have better brand recommendations.