A decent HD monitor will synch down to a lower resolution. For best results, stick to the same aspect ratio as the monitor itself is to avoid distortion.
Some HD sets are 16-10 ratio (1920x1200) and others are 16-9 (1920x1200, 1600x900).
Sometimes, they'll only expose a few or odd resolutions. 1366x768 is common. You should be able to see which resolutions are feasible because you'll be able to scale the desktop to them.
Actual performance needs vary by game and desired detail and if you want to run at exact resolution or scale up (if the interface is small, you might want to use a lower resolution for legibility, especially at TV viewing distance).