So, you want to root your U.S. or Canadian Galaxy S3 with the Qualcomm dual-core 1.5Ghz CPU?
UPDATE: Please see updated root method here instead!
If your Galaxy S3 model started with SGH-T999, SGH-I747, or SPH-L710, you can follow this step-by-step guide and root your phone within about 5 minutes. No, this will not erase any apps or settings on your phone and does not affect your phone in any way other than install ClockworkMod Recovery and give you root. (So if you have problems after rooting, don’t just assume it’s from rooting, it’s not.)
Works for all Gingerbread, Android 4.0.3/4.0.4 ICS, Android 4.3 Jelly Bean, and Android 4.4.2 KitKat! (For 4.3 and 4.4.2 please use the other SuperSU zip provided in Step 3.)