Motherboards will usually have some sort of power lights on them, but if even if the problem isn't the power supply, it never hurts to have a spare power supply you can use in your new machine. Since they are one of the most likely to fail components.
It's highly likely that there will be no visible signs of damage to the motherboard even if there are. Also there can often be a cascade of problems. IE. bad power supply fries motherboard that fries memory which might fry the motherboard if put in a new machine. I would just junk the ram from the dead machine if it turns out that the problem is not just a dead power supply.
I agree with everything Rackhir said.
The problem with motherboards is that they don't visibly show signs of what is wrong usually. A small short (usually caused by a power surge from a failed power supply to one of the supplies) with be nigh invisible. Really, the only visible sign of any circuit component failing is going to be a capacitor or inducer blowing, which will create scorch marks. If it's something like a resistor or a logic chip, there's no visible sign.
Merkuri said:
And now that both the place I bought it from and the manufacturer refuse to have anything to do with it I think I'll just throw out that cord. I have plenty of spares. I kept it with the idea that if somebody was going to replace my power supply they'd probably want to see the cord to "prove" that it was faulty. Even if it's not faulty, I don't want to risk destroying another (more expensive) power supply when I have plenty of cords that I know work.
No harm in that, though as Rackhir said, power cables are almost never the problem. The only things that can really go wrong with a power cable are a break in the wires (which would mean it wouldn't work at all) or a faulty ground, which is more likely to be wiring in your home. It's likely your power supply has actually been dying for some time now, and it's happenstance it happened when it did.
Before junking the whole machine though, here's what I would test, roughly in this order:
Replace the power supply. If your computer boots up, great! If not, you have a spare power supply, which is never a bad thing.
Take out all the RAM. Then put in one stick, and try booting up your computer. If it boots, it's probably a faulty stick of RAM - find out which one and throw it away (or replace it). RAM is fairly cheap, just make sure you get the right kind. If you have some spare RAM lying around you can use to test with too, that's even better - then you can rule out all the RAM being bad.
If you have a floppy drive, a copy of your BIOS and update utility, and are comfortable with it, I would try booting from the floppy and attempting to reinstall the BIOS. You can find specific instructions for your motherboard online - likely at the manufacturer's website, but also with Google as well.
If it doesn't boot from a floppy at this point, having tried the above two steps, there's nothing more to do.