After putting the bits of our future MythTV box back together, the system now powers up, but the SATA hard drive is not detected by the motherboard. Yawn. The SATA DVD drive, on the other hand, is recognised, so I tried the hard drive with that device's cables, but to no avail.
With no other SATA computer in the house, I have no idea whether it's a duff drive or some other, more obscure problem. There are plenty of stories out there about the Seagate ST3500641AS not working with Macs, due to the use of Spread Spectrum Clocking (SSC), but this isn't a Mac, of course.
It looks like I'm going to suffer another hardware-induced delay while I get this figured out.