A pre-rooted version of the new 126.96.36.199 Fire TV software update has been released by rbox. As always, this update is identical to the stock release from Amazon, but allows you to update without losing root. Go download it from the ROM page and install it on your rooted Amazon Fire TV to update to the latest and greatest version without losing root.
If you’re new to Fire TV rooting, check out my starters guide to learn how to use this custom ROM. This ROM will not help you root a Fire TV, since your Fire TV must already be rooted to install it.