Empire Earth is now the second game I've bought off GOG, both of which have had problems. Fallout 2 was just colour glitching which has now been sorted but EE just refuses to play. Whenever launching into anything after the menus the loading screen either freezes once it reaches 'Loading complete' or sometimes it actually loads and then freezes. It's quite infuriating, I was looking forward to playing this...
I should point out though that I'm not trying to troll or flame, I personally really like what GOG are doing, it's just a shame that I've seem to fallen under unfortunate circumstance, everything else about the service is excellent.

No one can help you unless you provide your system's specifications.

I recall having problems with EE1 because of my multicore processor, but I can't recall exactly what the problem was. It went away by just assigning only one core to the game.

Really, really sorry guys, totally my fault, seems in the video setting that I just had to change it from Direct3d to Direct3d with Transform and lighting, there were no multi-core issues though, I'm running an AMD Athlon x2 235e

Glad you sorted it out; your solution may be helpful to other people, too.