If you have been using IOroot and TowelRoot to root your LG G3, check out the new PurpleDrake rooting tool on XDA forum. Created by IOMonster, jcase and autoprime, PurpleDrake supports most of the LG G3 variants on T-Mobile (D851), AT&T (D850), international model (D855) and more!
Available for Linux, OSX and Windows, the rooting process with the PurpleDrake is pretty straight forward. You would need to first enable the USB debugging option on your phone, download and run the appropriate tool for your OS. Note that if you got the LG G3 from Verizon Wireless (VS985), IOroot is the only tool you can rely on now.
Not all LG G3 variants are tested yet, you can refer to this spread sheets on Google Drive for a detailed compatibility list.
Unfortunately, the rooting process will trigger LG’s built in root flag, which will affect your phone warranty when you need to send it to service center. This can be solved by restoring the root flag to its original status. Check out this post for the full instructions.