XDA Recognized Developer @geohot has developed a new root method that so far is working for many, many devices....Including ours!!! And believe me, it just does not get any easier than this!
Check out geohot's original thread here on XDA: I'm throwing a party and you should come **GS5 ROOT THREAD**
Don't forget to at least click his Thanks button, and consider donating to him if you can. There was a bounty started for rooting the Verizon and AT&T S5's, and they finally got their wish! And geohot gets to claim a nearly $19,000 bounty!!! Amazing stuff! For those who may not be familiar with his previous work, he was the first to jailbreak the iPhone/iPod, and if I am not mistaken he did the same with the PS3!
So anyway, you are here to root your device, no? I guess we should get on with it then!
This is confirmed to work on 4.3, NC2 firmware for the T999, as well as 4.3 and 4.4.2 KK firmware for the I747! It should work for the T999V firmware released this morning, and for the upcoming KK updates to the T999/T999L/T999N!
HOW TO ROOT WITH TOWELROOT
Developed By: geohot
Go to http://towelroot.com
Click on the red Lambda symbol to download the towelroot.apk
Install towelroot.apk on your device
Open the app
"make it ra1n"
You are now rooted! But need to follow a couple more steps.
Install Chainfire's SuperSU from the Play Store
Open SuperSu and let it update the su binary
When it asks if you would like to disable Knox, SAY YES!
Note: I tested this on stock unrooted 4.3 firmware. Build NC2. When I told SuperSu to disable Knox it just hung with the spinner going. I just had to close SuperSu, reopen and when prompted again, said yes and it worked just fine.
There you have it! The easiest root method we have ever had for our S3's! Enjoy!