10.1 nightlies are now officially supported for the standard definition Kindle Fires, So if you're curious about rooting and flashing ROMs to get a more stock Android experience, here are the official links for them, you'll still need to go about installing TWRP/CWR beforehand with the Kindle Utility that's over at XDA:
NOTE: Nightlies are not generally unstable. They might have a few quirks here and there, but it's mostly due to the CyanogenMod base being added onto every day. You could have something show up one day, and gone the next. Developers have to have drivers 100% ready before thy are able to get supported and listed on their repo.