The newest Amazon’s Kindle fire HD is shipping with locked boot-loaders. It would not prevent the devices to start with a method to root which already exist and go.
This method is confirmed for Fire HD 7, but you have to go on all new versions that are based on ICS.
First you need to download SU:
After you download the SU file, the next step is to extract the contents and pull SU from the next patch /system/bin/ and SuperUser.apk from /system/app/ and put them in the new directory from where you will be working.
After this step, the next step is to execute the following commands:
rm -r /data/local/tmp
ln -s /data/ /data/local/tmp
echo ‘ro.kernel.qemu=1’ > /data/local.prop
adb shell mount -o remount,rw /system
adb push su /system/xbin/su
chown 0.0 /system/xbin/su
chmod 06755 /system/xbin/su
adb install Superuser.apk
After you have executed all this commands, normally you should have the root access.
If you have any difficulty in the process, feel free to contact the support thread on RootzWiki