My PC:
Microsoft Windows XP Professional
Version 5.1.2600 Service Pack 2 Build 2600
System Manufacturer D845PT
System Model PT84510A
P4 1.8GHz
Total Physical Memory 1GB

Name NVIDIA GeForce 6600
Adapter RAM 256.00 MB
Installed Drivers nv4_disp.dll
Driver Version 6.14.11.6344

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The game plays fine single player, apart from a sound problem on my PC where the sound stuttered. To fix it I went to dxdiag (Start>Run>dxdiag) and reduced the sound from 'full acceleration' to 'standard acceleration'.

I then had a problem with multiplayer over my wireless LAN. Although my laptop saw the HOMM2 game I had set up on my desktop when I played HOMM2, in HOMM3 it just couldn't see it, even though they were the same PCs. Turning off the Windows firewall allowed HOMM3 on the laptop to see my PC again, so I knew that was part of the problem. In the end I just allowed 47624 TCP and UDP as exception ports in the Windows firewall. One entry for TCP, one for UDP. Then the game I hosted showed up on the laptop and we were able to play.