Here's a easy-to-use tool based on the method in this thread.
If you like this tool, please click the Thanks button!
Even better, see my signature.
DISCLAIMER: As with any rooting tool, there is some chance that something will go wrong. Use of this tool is at your own risk, and I will not be responsible if you brick your phone in the process. Although there have been zero reports of this tool breaking anyone's phone, don't run it if you aren't comfortable with the possibility of having something go wrong and having to re-Odin back to stock or worse.
- Install the USB drivers if you don't have them already: Verizon_Wireless_I535_GSIII_Samsung_USB_Driver_v1_ 4_6_0.exe
- Download the package and extract it somewhere (you'll need 7-Zip or some other modern compression tool) Make sure you extract all the files to a folder somewhere! Running it straight from your compression program probably won't work.
- On your phone, enable installation of third-party apps (Settings --> Security --> Unknown sources, near the bottom in the "Device administration" section)
- On your phone, enable USB debugging (Settings --> Developer options --> USB debugging)
- On your PC, make sure you don't have any other Android devices connected, any Android emulators running, or any Android tools like PdaNet running.
- Plug your phone into a USB port on your computer. For best results, use a port directly on the machine, and not a USB hub.
- Double-click RootDebugfs.bat and follow the directions on the screen. Your phone will reboot 3 or 4 times during the process; just leave it plugged in. To be safe, don't switch to any other applications while the root process is running.
DOWNLOAD GS3DebugFSRoot R2 FOR US GALAXY S III VARIANTS ONLY!!!
(Released July 24, 2012)
If anything goes wrong, post here and I'll see what I can do.
* miloj, for the debugfs root method for the Asus TF300T
* sparkym3, for his script for rooting the Asus Transformer Prime - this is a modified version of that script
* ChainsDD, for Superuser