Every single tablet or phone I've owned since the OG T-Mobile G1 has been rooted. I've even flashed rooted Android ROMs on an HP Touchpad. Today, because I'm bored, I am going to explain how you can take a $50 Amazon Fire tablet and turn it into a premium tablet running vanilla Android. The process can be intimidating for those people who do not know their way around a terminal emulator or a command prompt, but it the reward at the end will be a $50 premium Android tablet, or a nice paperweight if you fuck it up.
First, prepare your device. go into settings, into the About menu, and click the serial number seven times. This will unlock the Developer Options menu, which you will need to go into so that you can select "USB Debugging." Then, if you use Windows, download this file onto your computer, extract it, and run it. Once you have finished with that, you will need to install ADB and fastboot. You can get these utilities out of the Android SDK, or you can get them through some Linux software repos. Windows users can get ADB and fastboot here.
Before we start the process completely, you need several things: A copy of Cyanogenmod 12 for the Fire, Google Apps for your device, and SuperSU and the SU binaries, and finally TWRP. I would recommend putting everything but TWRP from that list in the root of your Fire's internal storage.
Next, open a command prompt or terminal emulator session, and plug your (powered on) Kindle in. Let the drivers install if they haven't yet*. Navigate to where ADB and fastboot are installed and run:
adb devices
If everything worked out, you should have a series of characters on your left, and the status on the right should say, "offline." Unlock your Fire and you should see a dialog box asking to allow USB debugging from your computer. Allow it. Run the ADB command again and the status should be different. The next command you will enter will be
adb reboot bootloader
Once the tablet has rebooted into fastboot mode, you will enter
fastboot boot <name of recovery image>.img
After that, you can either install Cyanogenmod. CM 12 gets installed first, then you can install Google Apps and SuperSU.
As a final note, I am not responsible for bricked devices, failed marriages, terrorism, alien invasions, nuclear war, the cancellation of Gotham, or any other possible consequences of following these instructions. If you need help, XDA Developers is a great place to go to find it.