disable all the fancy features from your graphic card. there are similar post here in the forum.

Disable Extended Display (for testing) and run the game solely on the 19" Screen.

Is it possible to play solely in software mode? Maybe it helps to disable the graphics acceleration?

I had this issue as well and managed to solve it. Roman's third link actually points to right solution. The problem is in fact with cards that have more then one GPU on them. Mine is, for example, ATI Radeon HD 5970 which has 2 GPU-s. Some old games, including JA2, have issues with those cards and in order to play them you need to disable second GPU. Disabling AI in Catalyst setting does the trick.
However, the problem is how to disable it since the option doesn't exist in Catalyst drivers anymore.
I use a freeware program that manages profiles for that purpose.

I use RadeonPro to do it. Supposedly you can do it manually too but I don't have knowhow myself. If you need instructions on how to set it up I'll try to do step by step. Btw, if you run into other older games that have strange behavior you may also try to disable the second gpu and see if it helps. I know that Space Empires V has white screen flicker if you don't turn off second gpu. Probably more games that have it, just didn't run into many.
Interestingly enough, DOSBox games seem to work fine with both gpu-s.

RadeonPro doesn't start. I get an error message saying "RadeonPro starup failure. Error: RadeonPro Support Service communication error. Start RadeonPro Support Service and try again." RadeonProSupport gives me upon execution the error: "Windows service start failure. Cannot start service from command line or debugger. A Windows service must first be istalled (using installutil.exe)..." Installutil.exe isn't in the folder though, just InstallUtil.InstallLog.

Is there a dedicated x64 version or what else could be the problem?

On my AMD card, I created a profile for JA2, in the gaming 3d application settings.
and the trick for the artifacts was turning off the morphological filtering.

I just uninstalled the latest drivers and installed Catalyst 12.1. I am missing the catalyst control center but the card is recognized and seems to work, the resolution is fine and JA2 works so far without those graphics problems.

Thanks guys, I appreciate it a lot.