Yeah, but stupid me decided it'd be a good idea to RAID0 the two drives on the motherboard, and if I'm to have any hope of getting that data back, I'm going to have to leave them as-is for the time being.

As it turns out, the floppy drive only screws up when I set one of the new hard drives as the master drive to boot from. This makes no sense at all. And I don't want it even thinking about touching the RAID0 set that's in there, so I didn't want that to be set as the primary boot drive. Fortunately, I've got an old SCSI card in the machine that I use for basically nothing (I really should just remove it) that I can set as the primary boot drive at least long enough to get Windows installed.