It was only a matter of time since Samsung already released the source code for the AT&T, T-Mobile, and Sprint versions of the Galaxy S III. That time has come as all three are now officially rooted. Special thanks have to go out to XDA member howtomen for the AT&T version and cyper_zero for the Sprint and T-Mobile version. Also dawgslayer69 posted that the Sprint method works with the T-Mobile version.
You will need the latest version of Odin to flash the kernel and ClockworkMod Recovery onto the device. All that’s left is to use ClockworkMod Revcovery to flash root. The only difference between the three different carrier versions is the files used, but interestingly enough, the T-Mobile and Sprint procedure is the same. Just hit the sources below for your appropriate model to get started.