For Android 4.3 or 4.4.2 KitKat, you can also use Towelroot, which does not require a computer!
UPDATE: Users have verified this root method also works for U.S. Cellular Galaxy S3, BootMobile.
If you have an international Galaxy S3 (non U.S. and Canada) with model number starting with GT-i9300, DO NOT FOLLOW THIS GUIDE but see How to Root Galaxy S3! instead.
This guide is for Windows, for Mac and Linux, please see this root method here.
Step 1. Connect your Galaxy S3 to your computer via a micro-USB cable and make sure USB PC connection is set to “Media device(MTP)”. If you didn’t change it, that’s the default settings on your Galaxy S3.