EA has confirmed that Mass Effect 3 does not accept Mass Effect 2 saves imported from the cloud. Local save files which have been moved across consoles via the cloud are also unusable.
Players hoping to transfer a ME2 save file via the cloud onto a new Xbox to play ME3 must transfer the save file back to the original hard drive. In order to use a ME2 save file, players must move the save file back onto the original Xbox 360, and then transfer it manually onto the new Xbox, either via USB or using a transfer cable. Local save files which have been moved across consoles via the cloud are also unusable. Therefore, players without their original hard drives will encounter some problems, as there is no way to recover your save file.
EA's statement says that it understands "this may be a great setback to those impacted who have been following the Mass Effect franchise over the years. We thank you for your patience."