Despite those lovely Microsoft people providing you with the ability to save clouds... sorry, save games to the "Cloud" on the Xbox 360, it appears that Bioware and Electronic Arts are not making use of that function with Mass Effect 3... yet. EA, for its part, does concede that "this may be a great setback to those impacted who have been following the Mass Effect franchise over the years."
The folks at EA are, however, currently aware of this problem. We assume that they'll still be aware of it in the future too. How do we know about the company's current awareness level? "We are currently aware of an issue that may prevent Xbox 360 users from importing their completed Mass Effect 2 save game files into Mass Effect 3," is a comment directly from EA, so that's how.
There is more, "While Mass Effect 3 supports cloud saving for new or successfully imported Mass Effect 2 saves, it does not support the importation of endgame saves directly from the cloud. Additionally, this can cause it to not recognize save files that have been transferred via cloud to a console other than the one where they originated."
Here's hoping...