Other posts seem to have similar problems with ATI based systems freezing. However, I thought I'd be as detailed about what's happening for me, in case it helps.
The game will start out just fine and I'd be able to get into a Sandbox game with no problem (haven't tried anything else yet). It plays great and there are no issues (Although I did have to reduce the resolution to the same one I use for Civ 4). At some point I'll play out several tactical battles. The battles play out just fine but at some point, when it switches to the battle result screen, the UI seems unresponsive. Occasionally, it will let me return to the map part of the game. The same thing occurs when you are in the Diplomacy screen trying to make a deal.
I switched from Full Screen to Windowed mode and the issue would still occur. If I exit and restart Elemental then it seems to freeze on the Startup movie. A reboot resolves the issue temporarily. I disabled the Startup movie and it will let me get past the intro/dragon screen; however, it freezes on the menu (the background either doesn't move at all, or for only the first second). Again, a reboot will get me into the game.
In the game, when in windowed mode, I run into the same problem. However, I can get past the issues in both the diplomacy and post battle screens by clicking on an appropriate button, bringing forward a window from another application (usually a browser) for 10 to 20 seconds and then bringing the Elemental window forward. It will then have advanced to the next screen (say stat distribution) or back to the map. Also, the issue does not occur after an autoresolved battle. The post battle screens never freeze as long as you never leave the map screen.
Well, that's about it. I have tried the various suggestions (delete the screen folder, disable atitask, etc...) but a reboot is the only fix that's worked. Also, at no time has the game crashed. Attached is the debug.err file which was too big for paste.bin.
[Edit- Still happening with 1.06]