Does a 120Hz display help with console gaming?

I heard from some person that you can only get 30fps out of a console(ps3 or 360). Is this true? I'm planning on getting the Alienware Optx monitor to play my 360 on which is a 120hz monitor. However I also heard that you must be hooked up via DVI not HDMI to view anything in 120hz. Is this true? And if it is then would using a DVI to HDMI converter work? Please explain to the best of your ability.