On Wednesday, the CyanogenMod team started rolling out CyanogenMod 11 nightlies for the T-Mobile, Sprint, Verizon and International LTE version of the Samsung Galaxy Note 3. Now remember that these are nightlies, so there's bound to be some bugs in there, which there normally are in CyanogenMod nightlies, even months after a new Android release. These ROMs for these variants of the Galaxy Note 3 weigh in at about 208MB in size. Also remember that you'll need to download the newest version of Gapps (CM's own package from Android File Host) to flash along with the CyanogenMod 11 build, or you'll boot up without the Play Store, or anything Google related really.
For those that aren't familiar with CyanogenMod, it's basically a third party ROM, which has plenty of customizations for you, and you can usually count on custom ROM developers to support your device long after the manufacturers and carriers will. Heck, they've got KitKat running on the HTC HD2 which was sold running Windows Mobile! CyanogenMod is a rather unique ROM as they went corporate last year and are basically looking to develop their own "skin" for their own hardware. Which they've just released GalleryNext a few days ago, and had their own camera app last year (before they went corporate) called Focal, which is now available in the Play Store. CyanogenMod supports a ton of devices, and that's probably an understatement.
If you're interested in downloading CyanogenMod and checking it out on your device, just hit the appropriate link above for your particular device of course. I'm sure they'll have more supported very soon. And well, we aren't really for sure when Samsung will be pushing out Android 4.4 for the Galaxy Note 3. But we have seen plenty of leaks lately, so it could be anytime soon. Although the carriers still need to certify it, which do take some time, unfortunately.