There are two ways to root your galaxy 3 on 2.2/2.1
Z4ROOT(2.1)
1.Download the attached z4root.apk and install it.
2.Enable USB Debugging
3.Go into z4root and click on Permanent Root.
Super One Click Root(2.2)
Make sure your phone drivers are installed on the computer (Samsung Kies). Then follow the steps:
1. Enable USB Debugging.
2.In Superoneclick 1.7, check regeagainstthecage then click on the 'Root' button. After a few seconds, it'll say 'Waiting for device'.