The HTC Desire series appears to be notoriously resistant to rooting. Rooting techniques for one device can be incompatible with most other devices. For example, while there are ample successful techniques for rooting the 820, none of them work for the 820s Dual SIM, despite all the similarities between the 2 models.
Kingo Android Root was the only thing capable of rooting the 820s Dual SIM. Unfortunately, as with most other rooting and hacking tools, it's available only for Windows, so the Linux geeks would have to seek help from the Windows guys.
Using the software is quite straightforward: make sure that you're connected to the internet, connect your phone to the computer via a USB cable (with USB Debugging enabled) and launch the tool. It will automatically detect the device model and give you a confirmation message. Just click the "ROOT" button, and the tool will start working, downloading the necessary dependencies. The rooting process should complete within a few minutes, at the end displaying a final confirmation message.
As the tool uses an on-the-fly custom dependency download, I was unable to track down the actual rooting method it had exploited. However it should probably be straightforward if you can check the temporary folders of the machine at the right time.
Unfortunately the superuser management app installed by the tool seems to be a bit crappy; some applications like ES File Explorer seem to be unable to access root privileges via it. However, the su binary installed by the tool works fine, so you can still access root privileges via the ADB shell or a terminal emulator app.