As usual, follow at your own risk. This tutorial is posted for information purposes only.
Step 1: Download
the insecure kernel (based on the XX NEE ALE8 firmware) from Chainfire.
Step 2: Download
ODIN 1.85, which we’ll be using to flash the insecure kernel.
Once you’ve download both, extract using the archiving tool of your choice. I use WinRar, but if you’ve done this before, you’ll probably have your utility of choice already installed.
After extracting the kernel, you’ll be left with a .tar file.
Place your device into Download Mode by pressing and holding the Volume Down + Power / Lock buttons simultaneously, and pressing Volume Up when prompted.
Run ODIN, connect your device to your PC via USB, and within ODIN, click on PDA and select the .tar file you extracted from the insecure kernel.
Click Start and the flashing process will begin.
Now you will need to "upgrade," as it were, from insecure to full root. To do this, you’ll need to boot your Galaxy S III, and enable USB Debugging by browsing to Settings > Applications > Development.
Your insecure root allows adb root commands, but won’t enable your apps to get superuser access; thus, you’ll need to install SuperSU.
Step 10: Download
the sgs3-root-install.zip file [XDA Developers link], which contains SuperSU v0.89, and extract it.
With adb running, double-click the install.bat, and after a reboot, your Samsung Galaxy S III will be fully rooted!