This method was first described in XDA forum by user howtomem and is tested on the Samsung Galaxy SIII sold by AT&T and Sprint, T-Mobile, Rogers. It was not tested on non carrier versions of the S3, so do not try it on any other Galaxy SIII versions. If your Samsung Galaxy SIII is from AT&T and you have managed to brick it, chances of getting a free replacement is from very slim to none. You have been warned.
I found the instructions by howtomem a bit convoluted, so wrote this guide to make it simpler.
1. Download the Samsung USB Drivers for At&T
2. Go to Settings->Developer Options->Enable “USB Debugging”
3. Now connect your AT&T Samsung Galaxy SIII to your computer and install the device drivers downloaded in step 1.
4. Download Clockwork TAR
5. Download Clockwork Mod Recovery Superuser v3.07 from the link at the end of this post. After you download the zip file to your computer copy in to the root directory/main directory of the internal storage of your Samsung Galaxy SIII.
6. Download Odin3 – v3.04 from the link at the end of the post
7. Extract Odin3 into a folder on your computer.
8. Unplug your phone from the computer and switch it off.
9. Now switch it on by pressing [Volume DOWN] + [Home] + [Power]. Keep pressing till a warning message appears. Now press [Volume UP]. A green Android will show up on the screen with a message saying “Downloading”
10. On your computer go to the directory where you extracted Odin3 in step 6 and run it.
11. Now connect your phone to the computer. Once you have connected, Odin3 should detect your phone and on the extreme left hand side of the Odin window, the COM box should turn blue.
12. In Odin 3 click on PDA and in the dialog box that pops out , go to the folder where you downloaded Clockwork.TAR in step 4 and select it. Make sure the [Auto Reboot] and [F. Reset Time] options are checked.
13. Now click [Start] in Odin. Your Samsung Galaxy S3 will reboot once the process is completed.
14. After your phone reboots completely, switch it off again. Now switch on your phone by pressing down [Volume UP] + [Home] + [Power] buttons together. Release the buttons once the screen is switched on.
15. You should see the recovery menu now. To navigate the menu use the [Volume UP] and [Volume DOWN] buttons. To select an option from the recovery menu, press the [Power] button.
16. On the recovery menu you see now, select [Install Zip from SD Card] > [Choose Zip from SD Card] option and select the Clockwork Mod Recovery v3.07 zip file you downloaded and copied to the phone storage in step 5.
17. Press the [Power] button to flash the recovery image.
18. After the flashing is complete, go back to the main menu using the Back arrow. On the main menu select [Reboot System Now].
19. Once completely booted go to Google Play and download the latest version of BusyBox from the store. Install it. It will ask you for root access, allow it.
20. Now go to the app drawer on your phone and run the Superuser app. On the top right hand corner of the screen there will be the settings icon, tap it.
21. In the settings menu under General section, tap “Su Binary”. In the screen that opens, select “Update”.
22. Congratulations you have rooted your Samsung Galaxy SIII from AT&T or Sprint and also have the latest version of SU.
PS: You have only rooted the Samsung Galaxy SIII SPH-L710, the ROM is still the same.