Note for those with nuke powers: This was published under the account "RighteousFunby" which is now more or less dead. This was its sole surviving writeup. Yes, it was my account.
OK, a few things to start with:

OK, that done? Good. Now, here's a guide.

Firstly, take the covers off your PC. Leave the front fascia on, obviously, but take every other panel off. Next, take out all of the PCI, AGP and ISA cards in the PC, unplug all of the IDE cables, all of the floppy cables, all of the memory...and unscrew all drives and set them down somewhere for later. Once everything is unplugged from the mobo, remove the power supply. Then, as a final step, remove the mobo. Unscrew it and pull it out.

If you've had your PC for a while then you'll notice a large amount of dust. A very large amount of dust. Get some toilet paper and wipe it all up. All of it. And get used to it.

Blow all the dust off the PCI, AGP and/or ISA cards, and all of the memory chips. Blow into the CPU fan and the fan on the PSU to get rid of dust. It's handy to have a bendy straw, or plain McDonalds straw to do this.

Wipe off all of the dust on the drives, and slowly but surely put back all of the components you took out. Seal the case up and (if you want to do this less often) use blanking plates and masking tape to seal up gaps in the case where dust can get in. Just make sure you cover no fans

Now for the keyboard. Use flathead screw drivers to prise off all the keys, then clean out all of the crap underneat them. Remember to use something like Notepad to find which keys go where. And yes, the space bar is an ass to get in again.

Clean out the mouse. Get all the black crap outta there

Wipe down the monitor with water or polish or something, and do the same for any other peripherals.

That really should be it...enjoy your clean computer, and if anyone knows any better ways to clean out a PC, post em below. Mac help would also be appreciated.
OK, UPDATE TIME! smartalix suggested not using straws because of the spit factor. Yeah, it's a problem, but a very insignificant one...spit dries :D He also suggests a mini vac. I have one, but it needs batteries, and I have a PC so that I can spend as little money as is possible, so the straw stays.

Also, arieh says that wiping boards causes a potential difference which can fry them. True, but if you're really that conscious, use a straw (ducks)...