Microsoft does two checks to make sure your stuff you purchased on the Market Place is being played by you. Anytime you get a new X-Box, you need to transfer the licenses. Unfortunately, this can only be done once a year.
The first check is your gamertag. Of course, when you purchase something, it's tied to your gamertag. This is verified when you log on Live. If you can't login into Live, the next check is the console itself. If your purchased the content with the current X-Box, it will allow you to access the content. If you purchased from another console, you won't be able to access it offline.
You have to actually do this on their website. I just wanted to throw that out there in case someone didn't know.