Tonya and I just moved, and although moving is a traumatic experience in the best of times, it gets hairy when you try to move computers and not have much down time. Most people probably aren't in quite the same situation we are, but I thought I'd pass on some of the strategies we employed to retain what little of our sanity remains after dealing with a sick cat who needs special food that makes our other cat throw up instantly.
Boxes -- By far the most important thing I can recommend is to save your original computer boxes, along with their original styrofoam packing material. Tonya would call me retentive about it, but I save the box for anything the size of a hard drive or larger (which includes the incredibly ungainly Telebit WorldBlazer, which is larger than our PowerBook 100). Even I draw the line at boxes for normal modems, trackballs, and Ethernet connectors, since you can pack them in a box with cables and other miscellaneous electronic gear. I realize that original boxes are bulky and hard to store, but they stack well in an attic (we're considering using them as room dividers due to a lack of attic space in otherwise spacious quarters) and enable you to avoid throwing out the styrofoam, which generally isn't easy to recycle. Whatever the pain of storage, it's worth it for the knowledge that your computer equipment (which may be among your most valuable possessions) is as safe as possible from the vagaries of the movers or from your well-meaning but butter-fingered friends.
Cables -- My next piece of advice concerns cables and may earn me some more retentive points. As long as you're dismantling everything and hopefully sorting through cluttered boxes of miscellaneous cables, take a few minutes to label and organize cables into basic categories. I now have plastic bags for cables and adapters related to power, SCSI, telephones, serial connections, and so on. I didn't worry so much about the organization before moving, because although my cable box was messy, I had a pretty good idea of where everything was. However, finding the cable box itself is hard enough among the detritus of the move, and trying to find a specific cable within that box would have been just about impossible without prior organization.
Speaking of cables, if you're used to networking two or more Macs together, consider visiting your local Homeowner Hell store and purchasing a few 50 and 25 foot lengths of 4-wire phone cable, along with several of those handy little connectors that let you join two cables into one. In a pinch, remember that PhoneNet connectors can also serve as temporary methods of patching together two phone cords (for either LocalTalk or the phone). In our case, we wanted to create an ad hoc LocalTalk network until we wired for Ethernet, and between that and some awkwardly located phone jacks, we used close to 200 feet of cable. It's not expensive and it never hurts to have it around.
My final cable comment is that you should think about how you will be getting power to your Mac or Macs. Often when you move, the power outlets aren't in quite the right spot, and you need to use an extension cord or power strip or something. We also strongly recommend that you consider getting a UPS (uninterruptible power supply) if the power in the new area is at all flaky. They're mildly expensive (several hundred dollars for a good one), but the peace of mind they provide is well worth it. If a UPS protects you from losing significant work just once, it has paid for itself, and it's even more worthwhile if it prevents a piece of more expensive equipment from being damaged when a tree branch hits a power line. (The electric company will reportedly often turn the power off, on, and off again in an attempt to burn the branch off the line, and one of our hard disks once fell victim to this practice.) We now have four UPSs, having just bought another one because the our SE/30 server can no longer share mine, being in a different room.
Furniture -- We took the opportunity of moving to refit our offices with good desks that can be adjusted to proper ergonomic heights. I've been working on a door slung across two cubbies since 1988, and although my sister and I chopped an inch and a half off the bottom of the cubbies three years ago when my hands were hurting badly, it still wasn't a good solution. After lusting for the pricey AnthroCart desks - which seem infinitely adjustable and sport more accessories than Barbie - we happened on excellent substitutes at the Swedish housewares and furniture store, IKEA. Our desks, made by a Swedish company called Jerker (I suspect it sounds better in Swedish), consist of two side supports with holes drilled every inch or so. They come with a desk platform and a monitor shelf standard, and you can flexibly adjust the height when you assemble them. The desk platform has two slide-out shelves on either side for holding papers or perhaps the mouse. Although not as easy to accessorize as the AnthroCart furniture, the Jerker desks can add an extension shelf that rises above the monitor shelf, and rotating platforms that attach to the side supports.
Tonya's desk doesn't have the extension shelf, and she placed her two rotating platforms on the right-hand side, the lower one for her LaserWriter Select 360, and the upper one for a Color OneScanner. I got the extension shelf, being taller, and put my two rotating platforms on either side for holding PowerBooks at typing height while standing up. The desks take an hour or so to put together with the included graphical instructions (which were much appreciated, since we don't know any Swedish), and seem to be extremely sturdy. The secondary reason for getting new desks is that we had no room in our old offices for some equipment we'd received several months ago as part of payment for a white paper about the Internet that I wrote for Apple. (Ask your dealer if you want a copy.) Now I'm up to using a 20" monitor as my main screen and a 15" as my secondary, and Tonya's using a 17" as her main screen and a 15" as her secondary (and Geoff uses a 15" and a 17" as his two monitors). Needless to say, we can't recommend multiple monitor setups too highly, and we consider them one of the most significant reasons to use a Mac over a PC.
Oh, the Jerker desks from IKEA cost about $240 for the desk, $30 for the extension shelf, and $15 each for the rotating platforms, which you can get with a large shelf (as we did), or a smaller shelf for holding a phone. They're pretty heavy, so shipping might be expensive, but if you have an IKEA store near you, check them out sometime.
Connectivity -- Finally, if you're as connected as we are, give some thought to your connectivity when you move. We went from a 56K frame relay Internet connection and two analog phone lines to a single analog line that we have to share for modem and voice use. We had hoped to have at least more analog lines in quickly, but due to lack of wires, U.S. West hasn't provided them just yet. So, to enable people to leave messages when the phone is busy (which is often, with two people using modems), we're thinking of having the phone company install voicemail until our other lines appear. Equally serious in many ways has been the slowdown in dealing with Internet email, and for that I set up AutoShare to reply to all my incoming messages. It tells people why my response time may be much slower than in the past and also provides them with email (<firstname.lastname@example.org>) and Web pointers to my Frequently Asked Question list, which answers many of the questions I get via email.
AutoShare is good about not sending replies to addresses that I've preset as being mailing lists, and it also only sends one response per person, even if they send me more than one message. AutoShare only works with Apple Internet Mail Server (or the older MailShare), but you may be able to simulate the same thing with your provider using the Vacation program. Not all providers will have it installed, but try using the Vacation command from the Unix prompt, or try MR Mac's VacationMail program, a shareware Macintosh program that communicates with your host to set up the Vacation program.
I can't pretend that this advice will make your move go smoothly, since that would violate a basic law of the universe. But if you can reduce the tension of fighting with your Mac and your connectivity during the move, that's a good thing and worth the attempt.