When you say "unplugged everything" does that include the internal storage devices? You might be able to at least get into the BIOS with them removed -- if so, the problem could be one of the drives dying.
No, it didn't. However, I just tried that and still no good. At that point the only thing plugged in was the GPU, RAM, CPU, and the various front headers. Thanks for the idea!
Still haven't heard back from Gigabyte support, but it IS still under warranty, as is the CPU. Just not sure how to tell for sure whether it is a problem with one or the other.