It just came to me I didn't present my solution here. Well, I got it working, finally. Oddly enough I had to use a D-SUB to HDMI converter (DeLock). The wiring went like:

1. DVI to D-SUB adapter
2. D-SUB to HDMI adapter
3. HDMI cable connected to video grabber
4. HDMI to DVI adapter
5. My A1082 Cinema Display through DVI cable

Weird, I know. Slightly crazy, I admit. But it worked!