Well it seems root methods are getting easier. JCase, a reputable Android exploit developer, and his team, released a one-click root method app for the LG G3 working on most US carriers (Verizon, AT&T, Sprint, and T-Mobile).
Stump-Root is simple. You install it, run it, reboot when it tells you, then install SuperSU from the Play Store. Enjoy.
Download:
Stump-Root 1.2.0Change Log:
v1.2.0
Rewrote Bruteforcer, time reduced from typical 1-2hour to 1-2minutes
Added support for older/other LG devices, like LG OG and LG GPad (all 2012-2014ish devices should be rootable with stump now)
Added a lot of devices to known offset list
v1.1.1
Added Support for LG-F400L Build/KVT49L.F400L10k
Added Support for LG-D851 Build/KVT49L.D85110c
Added Support for LG-D855 Build/KVT49L.A1407214586
Added Support for LG-D855 Build/KVT49L.A1406439589
Added Support for LG-D855 Build/KVT49L.A1407214586
Added Support for LG-D855 Build/KVT49L.A1405331481
Added Support for LG-D858 Build/KVT49L.D85810a
v1.1.0
Spelling corrections
Added Support for LG-D855 Build/KVT49L.A1407411578 (Some firmwares) - Yay whoever did this, first result from the bruteforcer!
v1.0.2
Added bruteforce mode for unsupported models
Added Support for Flex D959 TMobile
v1.0.1
Added patch detection
Added Support for D852 Bell 10B
Added Support for D852 Rogers 10B
Added Support for D852G Videotron 10C
v1.0.0 - initial release
Support for Sprint LG G3 release firmware
Support for AT&T LG G3 release firmware
Support for T-Mobile LG G3 release firmware
Support for Verizon LG G3 release firmware
Source: XDA
http://forum.xda-developers.com/lg-g3/orig-development/root-stump-root-lg-g3-sprint-verizon-t2850906Tested and working on Verizon LG G3