All three variants of the SGS III that are coming to the US have now been officially rooted. XDA-Developers members: howtomen, cypher_zero, and dawgslayer69 have all contributed to the AT&T, Sprint, and T-mobile versions respectively. Links are below.
Note: Rooting cam be a dangerous procedure if you do not know what it is or have done very minimal research. IT WILL VOID YOUR WARRANTY. You should check around the forums and understand all the internet lingo that the guys over at XDA use. Nandroid Backup, ROM, flashing and CWM (Clockwork mod) recovery are frequently used in rooting and ROM flashing. You should make an account with XDA (it takes 3 seconds, it’s free, and XDA has some of the smartest android developers that can help you in almost any situation.)