Log into your account at Xbox.com and do an Xbox 360 License Transfer
Online purchased 360 games and video have a sort of dual licensing. They are primarily bound to your account with a secondary binding to the machine. You can run the full version of these games on any machine while logged into the account with which you bought them and anyone logged into any account can run the full version of the games on the machine to which they're licensed. The license transfer process will license the new machine for the content, un-licensing the Xbox on which you bought the content. Once you finish the transfer, anyone who might have been using that content on the old machine using a different Xbox Live account will no longer be able to (well, they'll get demo functionality). If you sold the old machine that's probably not an issue.
There shouldn't be any problem downloading the content to a new machine using a Silver XBL account.