Actually my problem is with my new XB360 but the reason I posted here is because using esentially the same set up I have no obvious similar problem with any of my original Xboxes.
They're used via an official MS HD pack component cable going into the same (CRT at 480p) TV as my XB360. I've tested my XBoxes and XB360 at both 480p and 1080i
and the XB360 shows the interference at both resolutions. There's a hint of some sort of mild interference at 1080i on the Xbox most noticeable when using the menus in XBMC but it is no way as obvious as with the XB360.
Initially I thought the independent make XB360 component cable I was using might the problem. So I bought an official MS replacement but that resulted in no improvement.
I've had similar trouble on a Philips CRT when using a RGB SCART cable with a PSOne. That was solved by either re-tuning the TV to a blank channel before switching on the console or, as I eventually did, removing the video and audio output SCART pins.
You can't do the latter with component, obviously, but the the former ie,. re-tuning to a blank channel before switching over to whatever AV input you're using, might be worth trying.
I've also had intereference trouble when connecting my amp via the AV output block provide on many types of games console cables. This was resolved by a), using a better quality, shielded, cable to go to the amp and

. using a separate in line AV breakout connector instead of the the AV block,
In my case though the TV I'm now using isn't connected to anything else ie. no aeriel or cable TV it's just being used as a monitor for gaming/XBMC. I'm not online with it either so no problem for me from that quarter. My UK PAL Dreamcast suffered from some nasty inteference problems when connected online until I re-routed the connecting telephone cable away from the TV, other cables and replaced the telephone wall socket with a proper BT (British Telecom) one.
So where on earth is the interference coming from in this case? If it is a grounding issue as some others have suggested how (details please) do you go about resolving the problem?