Dodging can take some getting use to, but it's absolutely essential to master, especially if you find yourself w/o a vehicle on an Onslaught map.
There's no strafe/circle jumping here, but there is a similar movement in order to get across maps faster. You can't dodge multiple times in a row, you need a little bit of a "breather" between dodges, so what I do is basically start running forwards, then start looking a little to the left or right (so that I'm turning one of sides towards the direction in which I want to keep moving). Dodge towards the direction you want to be moving (remember to use the double jump to extend the distance of the dodge) and then move your view back to the forwards direction, always holding down the forwards key. A second or two later, turn sideways again, dodge-jump and then look forwards again.
Once you got that down, you can speed it up, so that in the end, the movements you're doing with your mouse and the keys are kind of similar to circle strafe jumping and on screen it looks like you're "snapping" the mouse to the right and left as you do your dodge jump. In reality, you're not snapping anything, and the movement has nothing to do with how fast you can go like it does in the Quake series. All you're doing is running forwards, turning sideways, dodge-jumping and turning forwards again in mid-air so you don't lose a beat in walking forwards.
In UT2003, you could also "dodge-boost jump". I never quite was able to figure out how to do this really well, but basically you'd power up the shield gun, aim it at the ground in front of you on an angle, then dodge backwards just before releasing the shield gun, throwing you a good few more meters then a regular dodge. Of course, you could extend that by jumping again at the height of your dodge.
Apparently you can't do this in UT2004, unless the UT2003 mutator is turned on. I've also noticed that shield gun jumping, period (UT's version of Quake's rocket jump) is no longer as effective as it was before. Anyone else notice this?
BTW, James, there's another movement tip you should know about: lift jumping. If you're on an elevator, just before it stops or slows down, you can jump and that will send you flying up into the air, the momentum of the lift added to your own jump. DM-Antulas is a good place to practice this.
(Sorry if you know this all already, I assumed you didn't based on your dislike of dodging)