Thanks to hoae.ucoz.net I was able to get my copy of HoAE running on Win7.
There is a number of solutions to this problem, and all are listed here -> http://hoae.ucoz.net/forum/2-33-1
I used the 4th fix, "Advanced 3d Patch", on my retail copy which works perfectly.
In more detail, I installed the fallowing patches:
1. 1.1 patch from http://www.heroesofae.com/en/?page=news_l&item=34
2. "3d Patch" from http://hoae.ucoz.net/load/patches/misc/heroes_of_annihilated_empires_3d_patch/6-1-0-16
3. "3d Patch with save fix & 2gb+ memory fix" from http://hoae.ucoz.net/load/patches/patches/heroes_of_annihilated_empires_3d_patch_with_save_fix_2gb_memory_fix/5-1-0-30
This should work on the STEAM version, although I have no way of testing that. You probably will have to skip the 1.1 patch.
Forgot to mention that the above will not fix the problems with in-game movies not playing properly. But there's a fix for that too!
Right click on your Heroes Of Annihilated Empires shortcut, select Properties.
Under "Compatibility" tab check off "Disable desktop composition".