There's been a bit of a kerfuffle with Cyanogen, Inc. today: conflicting interests in the Indian market mean that its relationship with OnePlus is probably over. But the development of the community-based ROM continues, and owners of the Verizon variant of the LG G3 can see for themselves tonight. The phone now has a nightly build of CyanogenMod 11 (Android 4.4) of its very own, and more are sure to follow soon.
The Verizon version of the G3 (model number vs985) is the first one to get CyanogenMod, made possible by the Bump! bootloader unlock tool and a stable version of the TWRP recovery. The various carrier and international models should follow at some point; CyanogenMod development usually relies on one of the team members owning or having access to the device, so it's probably pure chance that the Verizon version got the ROM before others. In the meantime you can check XDA and other user forums - there are already some AOSP and open-source CM builds floating around, including some Lollipop versions.
Frequent ROM flashers, you know the drill: the initial nightly ROM is bound to have some problems, so avoid it if you're bothered by a potentially less-than-stable system. The Verizon G3 should be included in the monthly M builds, if not next month then sometime soon after. If you've got your custom recovery ready to go, hit the source link for the download, and click here for a flashable Gapps ZIP for access to the Play Store and Google services.
Source: CyanogenMod download page