Xbox live account question?

I sold my Xbox while back ago and I recently bought a new with a blank slate. I downloaded my old profile where I bought several games on there. When I signed in the new Xbox the profile said I didn't have those games. I thought once you bought it, it went to your account so when you switch consoles you can retrieve the stuff you got on the orginal. Is there was to download these games again without having to pay for them again?