One of our favorite Android hackers, Justin Case (aka @TeamandIRC on Twitter) has finally posted his root method for the LG G3 and it does work on every variant of the LG G3. This includes the Verizon, AT&T, Sprint, T-Mobile and every other variant out there. It's actually really easy to root the LG G3 with Stump root actually.
All you need to do is download the Stump Root APK, install it like any other app. Once it's installed, go ahead and open up the application. You'll want to press the "Grind" button in the app to root your phone. Once that's done, all you need to do is reboot your device, and you're just about done. there's a few other things you'll need to do. Now you'll need to download SuperSU from the Play Store. Let it update it's binaries, and after a second reboot you're all set. However, I did not do a second reboot, so that isn't absolutely necessary, but it is recommended.
You'll also still need a custom recovery like TWRP. However TWRP and CWM are not yet available for the LG G3, at least at the time of writing this post. So unfortunately you've got root but no custom recovery. So if you're just rooting to use things like Xposed, Titanium Backup, etc., you're fine.
So that's how you root the LG G3 with Stump root. Pretty simple, although there's no TWRP yet