My understanding is that as long as you are using the same gamertag and xbox that you originally purchased it on you can just redownload it.
If you use it on another xbox it will be in a trial mode only unless you are logged into xbox live with the purchasing gamertag.
If your xbox breaks and you want to be able to use your downloads on a new console offline, you can use the new DRM tool on the xbox.com web site to change everything over to your new console.
*If you get your console replaced by Microsoft, they should automatically transfer the licenses over to the new system from now on*
(They didn't do this in the past, or they sometimes forgot to do it, which could be why the full version didn't come up unless you were on xbox live with your orig gamertag.)
The DRM tool is a new thing from microsoft, about a month old. Here is the link for the other poster to get his downloads licensed on his new console.
DRM tool:
http://www.xbox.com/en-US/support/systemuse/xbox360/licensemigration/default.htm
One other thing, if you use the same gamertag you originally did to purchase any xbox download (except rentals) it will not show a points number, only a check mark. If you try to purchase it again the screen will come up saying you previously purchased the content, and would you like to download it again. There is no charge, I have done this with some tv shows in the past; deleted them and redownloaded later without additional charge (on the same Xbox and Gamertag).