Can I use the Xbox 360 Headset for game sounds instead of voice chat?

I recently picked up a cheap-o Xbox 360 wired headset (one that plugs into the wireless XBox 360 controller I use). I'm not an Xbox Live Gold member, and so I don't need chat features or anything - mostly what I'm looking for is to be able to hear the game's audio with the volume on my TV turned down. However, the headset doesn't seem to emit any of the game sounds. So far I've only tried it with Fable II.

Is this a result of my stinginess? If I pick up a better headset will I be able to use it to listen to the game sounds? Or is this somehow related to the game - do only some games send game audio to the controller? Or am I crazy to think that I'll be able to hear game audio through a headset connected to my XBox 360 wireless controller?

Any help or advice appreciated!

Sorry to disappoint, but the headphone jack on the controller is for voice communications only. Game sounds, music, etc. are all played through the display device (tv/monitor).

To get the audio from the Xbox to your headset you'll need some sort of converter cable/connector that turns the red/white connector that goes to your tv to the jack that you have on your headset.

I ended up purchasing a surround sound headset to solve this particular issue.