Question: Is it possible to transfer my older content licenses to a new Xbox 360?

Answer: Yes it is. Microsoft provides a transfer tool on Xbox.com.

The license transfer tool on Xbox.com lets you consolidate all of your downloaded content licenses to one console. For example, if you downloaded items from Xbox LIVE® Marketplace on one console, then purchased another console later on and downloaded more content to the new one, your content would be licensed to two consoles. This tool lets you download your older content licenses to your new console.

For more detailed information on how to use this tool, go to the Xbox 360 Content License Restoration FAQ.  Also see:

http://www.xbox.com/en-US/Support/LicenseMigration/Home