UPDATE: This root method also has been verified to work with AT&T Galaxy S4 on MF3!
It involves an external microSD card that must be formatted in exFat format (you can do this easily on Windows by right-clicking and selecting format, you should be able to choose exFat as for format type), download and unzip the script, then copy all the files to your microSD card.
Download root script for VRUAME7 (Verizon)
Download root script for MF3 (AT&T)
Next, go ahead and download Terminal Emulator app from Play Store, open and type:
Once that’s done, simply download/install SuperSU from the Play Store, let it replace the Superuser app and also update the binaries. Reboot and you should have root!
Once you have rooted your Verizon Galaxy S4, you can follow Loki method to install CWM or TWRP recovery to install custom ROMs.
I should have an updated video tutorial but in the meanwhile, help yourself!