Luckly people at Android Police and Rootzwiki hand to us a solution, BurritoRoot tool, which is a new tool to root Kindle Fire 6.2.1.
And this sugar tool can be applied not also to the recent software model also to the older version. So if you want to root your Kindle Fire. The steps are given below.
Stuff you need.
How To Root Kindle Fire 6.2.1 Firmware.
Step 1: Plug your Kindle Fire with Computer.
Step 2: Open the root tool you have downloaded, and Root it.
Step 4: On command prompt, type the following:
Step 5: Use step command prompt again, type these lines:
adb push su /system/xbin/su
adb shell chown 0.0 /system/xbin/su
adb shell chmod 06755 /system/xbin/su
adb install com.noshufou.android.su-1.apk
Done you have succesfully rooted your Kindle Fire running on 6.2.1 firmware. Remember rooting your device will disable some features integrated with Amazon. But don't worry you can unroot with the same tool anytime.