There is a new Amazon Fire Utility available that will remove all of the Amazon branded apps and turn it into a fully functional tablet with Google Play. This tool is especially useful for people who want to purchase the Fire 7 for $49 and get more out of it.
There is no way to root the any of the new Fire tablets so the best we can do is some custom tools that developers release. Amazon Fire Utility is a relatively simple app that lets you Remove OTA, Remove Amazon Bloatware, Install Google Services and Chrome, Install Launcher hijack and Nova Launcher, Remove Lockscreen ads and Reboot Fire to Recovery.
This tool will basically remove all things Amazon, including the Kindle e-reading app, audiobooks, music, email, contacts, photos and any app you installed from the Amazon App Store. You are basically converting the tablet into a Vanilla Android device.
If you feel like experimenting with this new tool you can visit the official XDA thread, it has a bunch of feedback from the developer and comments from a bunch of happy users.