I have seen a couple of customer instances (including one at home 😉 where I wanted to move downloaded games and other content from an old Xbox 360 to a new system (as we did when we moved to an . Well, now there's a way to do it.
As the site notes, "With the license transfer tool, you can transfer the licenses of all your previously downloaded content from your old console to the new one."
Major Nelson has a How-To that walks you through the steps of transferring content licenses.
Also, as noted on the site, "if you received your new Xbox 360 as the result of a refurbishment or repair, the licenses have already been transferred as part of the repair process, and you do not need to use the license transfer tool, but you may need to download the content again. Check the repair and refurbishment instructions page for more information."
Also of interest, the Joystiq interview with Xbox Live GM Marc Whitten (thanks, Engadget).