When I follow links about rooting the Kindle Fire HD, I have the impression its not possible with the latest versions?
I have one bough just now, July 2013. Can I do it?
(The main issue is Google Play)
It seems that this is possible for both versions of the HD (7", OS version 7.4.3 and 8.9", OS version 8.4.3). I found XDA Developer forums for 7" HD version 7.4.3 and 8.9" HD version 8.4.3. For the purposes of making this post self-contained, I will assume that you have the 7" version.
The steps are:
Extract the file.
This step is very important. Uninstall any drivers you have for the Kindle Fire HD. Connect your Kindle to the PC via USB. Press the Windows Start button and search "Device Manager". Go to "Portable Devices". Right-click on Kindle and choose "Properties". Go to "Driver" and click "Uninstall". Then unplug the Kindle from the PC.
Download and install Kindle Fire ADB drivers. (see original forum for links to these)
On your Kindle Fire HD, go to Settings > Security. Mark "Enable ADB" - ON.
Connect the Kindle Fire HD to the PC via USB. Wait until automatic installation completes and Kindle is recognized.
Navigate to the folder where you extracted the Bin4ry method, double-click
RunMe.bat, and choose option 1.
On the Kindle Fire HD, select the option "Backup My Data" and wait until the backup completes (when the backup completes, press any key on the computer). Then select the option "Restore My Data" (This will not erase any data or files) and wait until the restore completes (when the restore completes, press any key on the computer).
Your device will reboot twice.
When your device boot, select the option "Restore My Data" once again (this will not erase any data or files) and wait until the restore completes (when the restore completes, press any key on the computer). Your device will reboot once again.
When your device boot, unlock your device and go to "Apps Drawer", you should see the "Superuser" App.