All is well again in my videogaming life: Microsoft sent me a new 360 (a refurb, I assume, though it seems to be in great condition). It came a lot faster than I expected, too -- in about five days (the customer service rep on the phone said that it would be 10-14 days.) Here's an excerpt from the letter they sent with the unit:

Dear Xbox Customer,

So that you may get back to playing and enjoying your Xbox quicker, we have replaced your Xbox console with a replacement unit.

Please accept our apologies for the inconvenience you have experienced by having to send your Xbox 360 in for repair.

Please accept this complimentary one month Xbox Live membership card as a gesture of our appreciation of you.

While I would have preferred my console didn't break in the first place, all-in-all the experience was pretty painless. They paid for shipping both ways (and provided a box to mail my unit in), the turnaround time was reasonable, and I got a free month of Xbox Live to make up for the time lost. Not bad.