While we have fheroes2 for Heroes of Might and Magic II, we also have VCMI for Heroes of Might and Magic III which also continues to advance as a free and open source cross-platform game engine reimplementation.
Recently the team behind VCMI released version 1.4.0 and 1.4.1 shortly after to fix up some issues. The list of what's new is…rather long. Sounds like some pretty necessary additions too including a high score screen, tracking of completed campaigns, VSync support, a tutorial, a video for winning / losing, smaller save files, mod compatibility checking when loading a save, simultaneous turns support for multiplayer, various AI improvements and so much more.
Ready for just one more turn?
You do need the original game for the data files which you can grab on GOG.com.