When sitting in a chair with your feet on the ground and your hands out, they should fall exactly over the keyboard without your wrists bending up or down. The desk should be at the correct height for this to occur. Note that the purpose of the ability to adjusting the height of a chair is so that your feet are properly on the floor and not for placing your arms at the correct height over the top of the desk.
When I was almost broke and living with several other guys, rather then buying desks, we built them. We went down to Home Depot and picked up some lumber and spent the next two days building our own custom desks. Because the desks were custom built I was able to place the top of my desk at my correct height. My hands rested correctly on top of the keyboard when it was on the top of the desk. The total cost of the desk was only $40 and a weekend's work. I doubt that I could ever find a better desk at a retail store. You don't need to build your own desk, but acquiring one that fits your body is essential! If your wrists are normally below the keyboard you might be able to get a keyboard tray to lower the keyboard's height. For the opposite, books or reams of paper can be used to raise the desk. Remember to try to find the best solution and not the cheapest/quickest one.
The Monitor Height
Keeping your arms straight is a whole lot easier if you have good posture. Part of this is sitting up straight. In my case when my arms are straight the desk is about two feet below my eye level. Because I don't have a three foot tall monitor (putting the center of the monitor at eye level) I now have a little wooden box that sits under my monitor that pushes it up six inches or so. This combined with the monitor's base was enough to raise the screen so I wouldn't slouch.
A Good Chair
Getting a good chair will help keep good posture. You don't need to spend $1000 getting the best chair on the market, but spending $5.95 wont do you any good either. You want a chair that has good support, fits your height and build, and the ability to be adjusted up and down. Your hands should be directly over the keyboard while your feet are firmly on the ground. Expect to spend around $40-$80 for this chair. It will be worth every penny in the long haul.
When I switched to a trackball mouse I saw a dramatic decrease in the amount of pain in my right wrist. Granted I hated those first two weeks when I had to learn how to use the mouse, but now I wont use anything other then a trackball. The reason is that when I use a trackball all of the movement is done by my thumb and not by my wrist. With traditional mice the wrists is constantly moving, often in quick small back and forth movements. The trackball was probably the best positive change I have made to help my hands. Bonus: For all of the gamers out there - How do you think those LPB turn around so fast?
I currently have a Kinesis Classic keyboard, but I didn't always have it. Over the years I have tried quite a number of different keyboards, from so called ergonomic keyboards, to one-hand keyboards and they are all now collecting dust in my closet. You can read up more about the Kinesis keyboard on Kinesis's site, but it all comes down to two things. The first one is that the pinkie (the weakest finger) no longer does a lot of work as the enter, space, delete, etc keys are in the middle where the thumb (the strongest finger) hits them. The second is the curved keyboard design helps to keep the wrist in line with the arm and prevents bending at the wrist. See if you can't find someone who has this keyboard and give it a try. It took me about two weeks to adjust to typing on this keyboard. An unanticipated side affect was that I could actually type faster (~130wpm) using this keyboard because the split keyboard forced me to properly type.
All of the above actions have lead to two main results for me. The first is that my wrist moves as little as possible. The second is that for the majority of the time my wrist and my arms are in a straight line and not bent. The combination of these two maintains a relaxed natural wrist position. This allows the tendons to work without dragging along the edges of the bone causing the inflammation and pain. I still have my wrist guards, but only wear them when I am working on someone else's computer. Some of the tips here are more expensive then others. However would you rather waste $400 buying new keyboards until you find the best, or would you rather have to go through surgery five years down the line? Surgery is just another form of the wrist guard, it will only remove the pain for a few years before it returns. Take a look at your computer area and see what you can change. Making simple changes to your computer area and remembering to take breaks from long periods at the computer (such as walking around and get some blood flowing) will go a long way. The past computer setup I had hurt my hands and I will pay for that the rest of my life. However I am not going to cause them further injury. The only times the pain comes back to my hands is when I am working with power tools for a long time, peeling a lot of potato's or some other repetitive task, but as soon as it starts to hurt I immediately stop and rest my wrists. My hands aren't pain free, but I can't remember the last time I had pain in my wrists because of my computer.
*I have never actually been diagnosed with having carpal tunnel, but I do believe that I was headed down that road.