This morning I checked the XBox service site for my repair status and found that my status had changed to this:

Repair Status: No pending repair

Could it be on its way?  Unfortunately there wasn't a new tracking number displayed or anything, nor do I know if I'll be notified via email of a new tracking number or anything.  I may be in the dark from now on.  And hopefully their site is not lying to me either!

Update

I knew they were lying!  I checked back and the repair status was back in its original state:

Device: Xbox360 Hardware
Serial Number: XXXXXXXXXXXX
Warranty Status: In Warranty
Repair Status: Device Received at Service Center