I pulled the drive out of the old machine and put it in the new one with no problems. Then I went to connect the cables and realized there was no data cable, just a power cable. I thought for a second and saw 3 ports on the motherboard that had the correct kind of connection, so I pulled the cable from the old machine and connected it. I wasn't sure which of the three to use, so I picked one at random.
I turned the computer on and ... the drive didn't show up. Then I remembered: BIOS. I fiddled with the bios, turning on each drive section one at a time until it saw the HD. It works!
My main thought when I was done? telepresence would be proud of me.