I had this game installed on my old system (running an XFX GeForge 8800GT), but I recently uplgraded my computer with:
Intel core i5 2500K
2x 4GB DDR3 G.Skill RipJaws RAM
MSi Radeon HD6950 2GB
among other things.

I just re-downloaded this, and when I started it up, the "intro" screens (where it shows company logo's and stuff) is just black (the sound plays and I can still skip through them).

The menu displays, but it 'flickers' a lot.

I tried starting a game, but once it loads up, the entire 'map' area of the screen is just black. The UI bars and minimap and stuff still show up, and I can click to 'move' (I can see myself moving on the minimap, so it's actually happening) but there is just nothing on the screen.

Could this be a compatability issue with an ATI card?

Do you have the latest DX9? (regardless of DirectX 11 that comes with Vista/7)

I'm pretty sure I do (although not 100%.)

Just had a little more look into it, and apparently the AMD Catalyst drivers don't work with this.

Supposedly it worked fine with 10.7, but 10.8 onwards have had this flickering problem (Although, the videos i've seen of this have only flickered, none have my problem of the blank screen ingame.

The problem here is that People on the HD6xxx cards can't use the drivers early enough for this to work. The first time I ever buy an AMD video card and this is what happens. Sigh.
Another update: I went into the Catalyst Control Center, and under "Gaming > 3D Application Settings" I unchecked the box "Enable Surface Format Optimization" under "Catalsyt A.I." and I can now see the intro screens and ingame shows up. Unfortunately the flickering is still there.

From what I'm reading, there was no consistent solution found except rolling back to 10.7.
I hope someone with ATI card will see this topic, all mine are NV and I won't be able to help much.

Have you tried disabling Anti Aliasing in the game options?

Kind of strangely.. when I disabled Anti-Aliasing ingame, the flickering went on for another minute or so, then seemed to behave itself.

Odd?

Yeah that's what I did on mine Seltox. The main menu and loading screen flicker, but once it gets in game it runs fine. Supposedly the next catalyst hitting this month is supposed to fix the problem completely (it's in the patch notes).

Unfortunately it's still flicker city with the 11.6 for me.

Found in Catalyst 11.6 release notes:

...
The game Kings Bounty now renders correctly.
...

For the blank video and map bug.