Rooting the Sprint Samsung Galaxy SIII is not all that bad, given you have the right tools and follow the correct steps. Some new updates have made it a bit different to root the device but not that difficult
Before you start:
you're going to need Odin 3.07, Team Epic Root From Recovery a .tar of the most update custom recovery tool. I prefer TWRP, Ill be using it in this guide but CWM wouldn't be that much different.
On the Sprint Samsung Galaxy SIII the bootloader has come unlocked so that will save some time, effort, and headaches
Preparing your computer
1.) Create a folder on the root of your C:\ drive named Root GS3
* this will just makes things easier later
2.) download Odin from the link above and extract it to your Root GS3 folder
3.) download the .tar of the recovery from the link above DO NOT UNZIP but copy it the same ROOT GS3 folder
Preparing your phone
1.) download the Team Epic Root from Recovery from the link above
2.) connect you phone to your computer in mtp mode
3.) copy the Team Epic Root .zip from to the root of phones storage or external sd card
Let's Begin
1.) connect you phone to your computer in mtp mode
2.) copy the Team Epic Root .zip from to the root of phones storage or external sd card, if you want to put into folder for organization that's on you.
3.) disconnect your phone from the computer and power it down, remove battery cover for ease of access later
4.) we need to get the phone into download mode, hold down the power, volume down, and home key until you get a warning. then the volume up key to continue
5.) connect your phone to the computer using the usb cable
6.) launch odin 3.07 from the extracted folder
7.) you will see com area all the way to the left turn blue and an added line in the log
*if you do not see this disconnect and reconnect your phone until you do.
8.) click on PDA module and select the .tar of the recovery img
9.) uncheck auto-boot, and f-reset time
10.) when odin looks like it does in the attached file click start
11.) when odin completes successfully you will have a green pass block all the way to the left.
12.) pull the battery and disconnect the cable on your phone
13.) no we need to get to recovery mode
14.) Re-insert battery, hold the volume up, power, and home key until you see the phone begin to boot and blue text in the top left corner then let go. if you miss the timing the phone will boot normally and you'll have to try again.
15.) on TWRP go to install
16.) it will point to install card by default you can pick external depending on where you have saved the root from recovery .zip to
17.) select the .zip file and swipe to install
18.) wipe cache/dalvik after
19.) then reboot system
20.) At this point, provided everything completed successfully, You are ROOT