I’ll be moving shortly, and will be taking a couple desktop computers. What should I do to prepare them for the move? Or can I just put them in a box and go? Thanks.
Basically, put 'em in a box and go. If you’re packing to be moved by yourself, just know what boxes have PCs and be careful with them. Wrap them up in bubblewrap or some blankets if going by movers.
It’s been about a decade since hard drives needed to be “parked” before moving.
If you’re paranoid about such things, back up your data to CD before moving. When it comes to keeping your files safe, it’s not possible to be over-paranoid. 
When at the new place, open up the PCs before turning them on to make sure no cables or boards wiggled loose in transit.
If you’ve got the old five-and-a-quarter inch floppies you have to insert a thin cardboard protector in the drives. That’s unlikely these days. I don’t know if LS-120 or ZIP drives have similar requirements.
Otherwise, just pack carefully, avoid shocks. The hard drives are pretty rugged but you still don’t want to drop it or hit it too sharply. Those (and the obvious monitor screen) are your most fragile parts.
To be safe, I’d take the hard drive out and hand-carry it. The rest of the components are usually fairly shock absorbant, and standard packaging materials should cover you.
-lv
I’ve had to move mine several times at Uni’ as long as its packed to withsatnd the odd pothole while I’m driving its fine.
I’d be more concerned about pulling loose leads and wrapping them up in the box. For some reason modem leads on all the new computers I’ve helped set up have all been of the type that plugs in and stays into the modem, I’ve had to chop off the little plastic bit that catches to package up the computer.
I always make backups when I move mine. If you’re super-paranoid, a Mailboxes Etc. can pack them for you, and they use good boxes and styrofoam. Just see if you can pay them to pack it and then pick it back up. I usually just put my tower in a blanket and wedge it between some stuff that’s not going to move in the car.
Keep ALL the cables, accessories, CDs and everything to do with the computer in the same box(es). It’s no fun when you try to set it up again and a critical cable has been packed in the box with your old yearbooks and said box is buried somewhere in the basement.
Ask a local ice cream store for the boxes that their cones come in as they are packed in a real nice soft foam that you can use.
Protect the monitor screen with something so that nothing touches it, perhaps put a tshirt or sweatshirt around the whole thing.
I too would recommend making a backup, mostly to protect against loss or theft of the computer. (So don’t put the backup disks in the same box!) Disk drive failure seems rare these days. I usually use a sheet of styrofoam as padding on all sides, but crumpled newspapers work fine too.
I wouldn’t remove the hard drive for shipping. IMHO it’ll cause more risks than it prevents. (You might drop the drive while removing it, or you might zap the motherboard with static, etc.)
In the future (it may be too late now) keep the box that the case came in. It’s the perfect way to move a PC, complete with styrofaom.
You can always get boxes from a computer store.
I’ll second LordVor’s advice. Just make sure you clear your hands of static before you handle the hard disk(s). Preferably keep the hard disk(s) in an anti-static bag. You can then further protect it with bubble wrap or even a towel.
The case makes an excellent hard disk protector. Don’t bother taking it out. You are far, far more likely to damage it taking it out, carrying it around and putting it back in.
I 2nd the suggestion to reseat all the cards (including memory) after the move.
The main issue is with the other stuff. Monitors are best moved in their original boxes. Reverse the unpacking instructions for your printer. (Assuming you kept the instructions, and the little plastic or cardboard shims/pins.) Be careful in particular with the ink cartridges, easy to see why.
You should wrap stuff in bubble wrap and then put tape all around that. Taping something shut and then wrapping will reveal itself A Bad Idea when you get to the other end.