My camera is fixed!
I've been thinking very bad things about the people at Fujifilm for most of last night and today but it turns out that the batteries that were new (bought cheap ones to ship with the camera) when I sent it off are mostly dead now. I figured I'd make sure I had good batteries in before I cursed some poor CSR out, so I brought some to rechargables with me to work. They've been plugged in all day. I just put in the freshly charged batteries and it works.
(Sorry for the profanity this morning on Gtalk.)