I know that the HD Homerun can be integrated into Windows Media Center and Media Portal. I believe it can also be used with Myth, but I'm not sure about that. Perhaps someone who has succeeded in using other software will chime in
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I'm using a HDHomeRun Dual that I got about a month or two ago with Mythbuntu running on a dedicated backend server, and a MythTV frontend on another box that is connected to my TV (in a different part of the house). It's a bit of a bear to set up (and even moreso if you cannot utilize or don't want to pay for the TV scheduling service that they push really hard to U.S. residents, but that won't even work for people in other parts of the world), however it's not
that difficult, and once it is set up it works VERY well. And if you use the MythTV backend you can either use their frontend software or you can use the new (still in Beta) version of XBMC "Frodo", which now has PVR support.
The software included with the HDHomeRun is good for setup and initial testing but to get the full power you will want to run some sort of software that lets you record programs. If you are a bit of a Linux geek, the easiest software to set up and get running is a command-line program written in Python called HDHomeRun Recorder, which I found out about from an article
here. That program does not have the nice GUI schedule grid, however it was easy to set up and opened my eyes to the potential of this device. But MythTV offers so much more control, including the ability to record from both tuners simultaneously. XBMC Frodo also has many of the major features of the MythTV frontend, but since it is still in Beta I'm not yet running it on my HTPC.
For those that don't get the backend/frontend concept (which I had a bit of difficulty understanding), the backend is basically what talks to all your tuners and acts as your PVR and scheduler. The frontend(s) (and you can have more than one) connect to the backend to obtain the content, and present the user-facing interface. So, for example, you can schedule a recording on any frontend you have (or even via a web-based interface) but the scheduler itself is on the backend, so if someone sets up a recording via any frontend, any other frontend will be able to see that it is scheduled or watch it after it's recorded. You CAN run both the backend and a frontend on the same machine (assuming it has enough CPU power to handle both functions) and many people do that, but I have not done that so far. You can also have "slave" backends but I definitely have not attempted that!
There are a lot of features in MythTV that I did not expect and that surprised me, so if you have a spare machine that isn't being used for anything else and isn't ancient, you might want to give Mythbuntu a try, unless you are pretty confident in your Linux skills and would prefer to try it on some other distro.
It was the fact that the HDHomeRun and MythTV worked together so well that caused me to start looking for a DVB-S2 device that might also work with MythTV (that and the fact that my satellite receiver died on me). If this new TBS box has DiSEqC and 22 kHz tone switch support I will be very interested. If not, then not so much.