Two years ago my first 360 had a RRoD. I bought a new 360 and transferred the contents of the old hard drive to the new one. I did a license transfer, redownloaded content and everything was fine.
I just moved to a new house that does not yet have internet access. I CAN play all of the games on demand and access any dlc content that was on the OLD 360. However, I CANNOT play any games or use any dlc purchased in the past two years on the NEW 360. I am using the same account I've had since my first Xbox.