Microsoft has released a workaround that allows Xbox 360 owners to transfer licenses, i.e. Xbox Live Arcade games and other downloadable content from the Marketplace (the exception being rented movies, make sure to watch them first), from your old Xbox 360 to a new one.
It used to be that whatever you bought on Xbox Live and downloaded to your 360, was attached to THAT specific console, as opposed to being attached to your Xbox Live account. This meant that if you got another Xbox 360, the only way to access your downloadable content from your other system was to log onto Xbox Live with the GamerTag that originally purchased that content.
Now Microsoft has released said workaround that allows you to transfer your files from one system to another (though you will still have to redownload them to actually get them on the system). To do so you must be able to connect to a computer that has Internet access and then sign onto Xbox Live with your 360.
The catch is that this process can only be performed once every 12 months. And Microsoft has reiterated that it’s only necessary for those people who have purchased a new Xbox 360 console, and isn’t necessary for repaired or refurbished systems.
Watch the video demonstration License Transfer Tool to see exactly how to do it: