[Fire HD 8 and HD 10 General] bibikalka:
Update - March 23rd, 2019.

There is now an excellent offline rooting method for HD10 (2017), HD8(2016-2018), all current FireOS versions, thanks to the fantastic effort by [MENTION=8132642]diplomatic[/MENTION] - link. This new root is obtained within a few seconds, so it is very fast. To install permanent root after mtk-su for HD10 (2017), HD8 (2016-2017), use the scripts by [MENTION=8978978]Rortiz2[/MENTION] in here: v2.1, or v1.0 (see this link for a Linux script). For HD8 (2018), there is a dedicated full bootloader unlocking and permanent root guide - link. For HD10 (2017) there is also a full bootloader unlocking procedure available - link. The historical Kingoroot rooting procedure for HD10 (2017) is below. As a friendly reminder, once you have a root shell (obtained by any means), ensure that you disable OTA updates as to avoid getting an unrootable update (except for HD8 (2018) - do not run the commands below before you unlock the bootloader!!!):

mount -w -o remount /system
mv /system/priv-app/DeviceSoftwareOTA/DeviceSoftwareOTA.apk /system/priv-app/DeviceSoftwareOTA/DeviceSoftwareOTA.apk_
ls -l /system/priv-app/DeviceSoftwareOTA/

The original Kingoroot method for HD10 (2017), all current FireOS versions - EASY SuperSu

Update v0.9, 12/29/18 As empasized by shonkin in this post, the method still works for FireOS 5.6.3.4. Enjoy!!! I would never have expected the hole to last his long ...

Update v0.8, 01/26/18 - as first reported by [MENTION=5769394]najoor[/MENTION] in this post, FireOS 5.6.0.1 is rootable! Today I verified this myself after a somewhat unsuccessful FlashFire update.

Update v0.7, 01/14/18 - [MENTION=6316731]freaky2xd[/MENTION] made a video with these rooting instructions, please follow it if you prefer a visual guide (here or here)

Update v0.6, 01/09/18 - there were reports that dr.fone app seems to be able to root the device as well. I took dr.fone for a spin, and based on its bloatedness and a few other annoying features, my personal recommendation is that you stick with the devil we know - Kingoroot (link)

Update v0.5, 01/04/18 - add a DOS bat file to remove any possible updates to Amazon packages

Update v0.4, 01/02/18 - title update

Update v0.3, 12/31/17 - light clean up; I got into a bootloop, and had to sideload a stock ROM & re-root - a.k.a. "eat my own dogfood" :D - Everything works fine.

Update v0.2, 12/30/17 - some redundant commands are removed.

Update v0.1, 12/30/17 - the rooting procedure is essentially taken from [MENTION=3497316]retyre[/MENTION] (here and here). Except, the instructions below include a lot of details, and handle mostly everything from the PC/ADB side. Try, and report back. GOOD LUCK!

Here is the guide to the painless root (while out of the box FireOS allows it; right now all FireOS versions up to and including 5.6.0.1 - the current OTA - are rootable). The key enabler is the original post by [MENTION=8784194]ztrund[/MENTION] (link), great work blazing the trail (and motivating me to get another Fire tablet ASAP, LOL).

Given that the devices will be shipping with the (older) rootable FireOS for the next few months (but beware of the upcoming updates - see below !!!), there is a good window of opportunity to acquire a rootable Fire HD 10, and root it. As of Dec 27th 2017, there are no reports yet of non-rootable OTAs, but those OTAs will be coming soon, count on it!

The utmost goal here is to preserve the earliest FireOS version that you get, and not let it get updated by Amazon on a whim.

I am starting with a recently bought Fire HD 10 2017 (light refurb from eBay, missed the Black Friday mega sale :( ). I have FireOS 5.5.0.0 (earlier than 5.6.0.0!), version name 5.3.5.1 (591450020)

Part I (avoiding Amazon updating procedure upon the initial Fire setup) - this can (almost safely) be skipped today (end of Dec, 2017), since there are no reports of unrootable OTAs yet

Low tech way (thanks to [MENTION=2966179]Blaiser47[/MENTION] and [MENTION=3497316]retyre[/MENTION] for suggestions!):

Unpack Fire HD 10, turn it on, it will immediately demand a WiFi access

Choose any option on that WiFi screen, press cancel, and then skip

Once Alexa shows up, swipe down from the top, turn on Airplane mode just to be sure

High tech way:

Set up a dedicated slow router, limit upload/download speed to ~25 kbps (this is the trickiest part, I have a dedicated Tomato router which I use to control traffic)

Unpack Fire HD 10, turn it on, it will immediately demand a WiFi access

Connect Fire to your slow router

Once Fire finds Internet, it'll immediately have "Checking for updates" on the screen, this is where the slow router should kick in, and do the trick of forcing the update to give up

Wait a bit for updates, hopefully, it won't find them, if found something, do factory reset, and repeat (on my 1st try it did find the update, although, could not download it quickly enough, I did a factory reset via Pwr&Vol+ recovery mode, and tried again - the 2nd time it skipped the update due to the slowness of the connection)

Sign in to Amazon account when prompted

Once Alexa shows up, swipe down from the top, turn on Airplane mode - no more risk of updates!!!

Part II (rooting via Kingoroot, disabling OTA, and getting SuperSu replacement, as per [MENTION=3497316]retyre[/MENTION] recommendations)

Take your Fire HD as is, do not do anything dramatic such as "factory reset", Amazon ROM sideloading, etc

Swipe down from the top, turn on Airplane mode - to ensure that there are no OTA updates during the procedure

In "Settings/Device Options", tap "Serial Number" 7 times, a menu "Developer Options" will appear

In "Settings/Device Options/Developer Options", turn ADB debugging to ON (under "Debugging")

In "Settings/Security", turn "Apps from Unknown Sources" to ON

Download ADB to your PC (link)

Setup ADB drivers on your PC, connect Fire to your PC, make sure "adb devices" command shows your Fire device, authorize ADB connection on Fire

Download SuperSu 2.79 (this exact version!!!) to your PC from this link, place it into your ADB PC folder. The filename of this apk will be assumed to be SuperSU-v2.79-20161205182033.apk below

Download the attached su64.zip to your PC (see the attachment below), unzip to your ADB PC folder

Open a CMD window in ADB PC folder (this will be called ADB_cmd window in the following steps), type

adb devices
adb uninstall eu.chainfire.supersu

This is a clean up of possible old SuperSu (just in case), ignore any errors you may get (if SuperSu is absent ...)

Download Kingoroot to your PC (link), install, let it update

Connect Fire to your PC; launch Kingoroot on PC; before pushing the big "ROOT" button in Kingoroot, uncheck a small box in the lower left corner for "Install recommended app" ; push "ROOT" button; wait for Kingoroot to root

Once Kingoroot succeeds, open a 2nd CMD window in ADB PC folder to your Fire (this will be called ADB_root window in the following steps), get a root shell, and disable OTA updates

adb shell
su
mount -w -o remount /system
mv /system/priv-app/DeviceSoftwareOTA/DeviceSoftwareOTA.apk /system/priv-app/DeviceSoftwareOTA/DeviceSoftwareOTA.apk_
ls -l /system/priv-app/DeviceSoftwareOTA/

ignore any errors you may get while doing this; after 'su', you should see root (#) prompt here

Switch to the first ADB_cmd window, type

adb uninstall com.lionmobi.powerclean
adb uninstall com.kingoapp.link
adb uninstall kingoroot.supersu
adb install SuperSU-v2.79-20161205182033.apk
adb shell "am start -n eu.chainfire.supersu/eu.chainfire.supersu.MainActivity"

Skip this step - it is not needed

Switch to the second ADB_root window (with # prompt), type

cd /data/local/tmp
mount -w -o remount /system
cp ./su64 /system/xbin/daemonsu
chmod 0755 /system/xbin/daemonsu
daemonsu -d &
cp ./su64 /system/xbin/su
chmod 0755 /system/xbin/su
am start -n eu.chainfire.supersu/eu.chainfire.supersu.MainActivity

On your Fire, SuperSu should pop up. Update SuperSu binary as "Normal", it should report "Installation failed." Proceed to reboot. (If it doesn't report an outcome ("failed") in a couple of minutes, go to the Fire's Apps and force-stop SuperSU and retry.)

Upon reboot, SuperSU should be functional. Choose "Grant" as the default access.

Uninstall all the junk from Kingoroot on your Fire, thanks to [MENTION=7298117]fstanis[/MENTION] for detailed instructions (copied here, executed from PC):

adb uninstall com.nemo.vidmate
adb shell rm -rf /sdcard/VidMate
adb shell rm -rf /sdcard/.a
adb shell rm -rf /sdcard/.DataStorage
adb shell rm -rf /sdcard/.UTSystemConfig

Some troubleshooting options:

If you believe you have the correct FireOS, but Kingoroot (or SuperSu) still fail, download the attached no_amzn_updates.zip to your PC, unzip to your ADB PC folder, open a CMD window in ADB PC folder, and type

.\no_amzn_updates.bat

The script will attempt to uninstall any apk updates to the official Amazon packages. Then repeat the rooting procedure from Step 1 skipping as necessary. See this post for more additional info on what the script does.

If you mess up your /system too much and get into a bootloop with "Fire" logo - use this post for links to the official Amazon ROM files; these bin's can be sideloaded via "adb sideload" in recovery

Want to say thanks by clicking the "Thanks" button ? ;)