I don't have a spare CPU cooler that will fit into my new motherboard. The one from my old motherboard only has 3 pins on the end of the cable (rather than 4), and the one from the dead motherboard has the screws on the heatsink spaced too far apart to fit the holes (which is bizarre, since they're both socket 775).
So, here's my plan: leave the existing heat sink and fan in place, but don't plug it in. Meanwhile, plug the giant heatsink/fan into the new motherboard, and go into the BIOS to see what it says. If it displays a speed then I know that the motherboard is fine, and the new cooler is broken. If I don't see a speed, I need to send the motherboard back. The only slight drawback is that the CPU will immediately start to heat up. I've done some digging on the web, and various random people suggest that it will be ok for a few seconds, and things will only start melting after a few minutes. So, if I'm quick enough, I should be fine.
I realise that this may not be the most prudent course of action, so I'll sleep on it before I implement it. If you see a tower of flame coming from my flat tomorrow evening, that probably means that things didn't go according to plan.