I just bought HoMM3 Complete and successfully installed it and run it on Ubuntu 12.04 using PlayOnLinux 4.2.11.
The game apperars to start and run fine (I was even able to complete a couple of turns and a battle) but on every game I've played, after only a few minutes into the game either the screen freezes or displays some weird pattern of relocated rectangular areas of the screen or the mouse becomes jittery. End result is that I have to quit as the game becomes unplayable at this point.

I attach a photo of my screen which I took with my phone. I had to crop it as the forum won't allow me to upload an attachment larger than 500KB.

Note that I haven't installed "HoMM3 HD Latest.exe" which, I understand, is used to support higher resolutions as I don't really mind the original resolution tha game shipped with. So the only thing I've installed is "HoMM3 Complete" as bought from gog.

Finally, even though the music was initially fine, it has now started playing at 3X speed.

Do people have success with playing HoMM3 Complete on Ubuntu using PlayOnLinux ?

Yes, I am able to play it on Wine! The screen does freeze an awfull lot and I hit F4 on average once every minute but I can live with that! I wonder why PlayOnLinux is touted as a superior alternative to Wine ...

That seems like an extremely frequent problem. I haven't had it happen in several weeks, but even when it happened to me regularly, it was no more than once every play hour or so.

Did you use taskset to bind HoMM3 to a single core? I don't know if that is related, but it has not caused me any problems, and various vague posts suggested using it would be safer than not using it.

PlayOnLinux is supposed to apply workarounds for known issues. It seems in this case whatever changes it applied made things worse, not better. For some games, it might be useful. For the games I play, I've never needed it.