I currently have two desktop computers up and running at home. My new one, P4 3.0GHz with 512MB RAM, and my old one (which my son uses), a P3 866MHz. World Wind runs on both machines. A little slower on my old machine, but fast enough. There is a problem, however.
On my new machine I run into problems while tilting the display to see 3D terrain features (I don't have this problem in Google Earth). My Old machine does this just fine. The difference is my old machine has a dedicated graphics card and my new machine uses Intel Extreme (or some such) which is on the motherboard and has no dedicated video RAM.
If you are going to be tilting the Earth (which is really cool) you must have a video card. If you don't have a video card there is still a lot you can do, however. You can check to see if your card is compatible at the World Wind Wiki.
If you're not sure what video adapter you have you should download a system information utility. I really like Belarc Advisor, it tells you everything including which if any windows updates you may still need to install.