"ATI cards of the 3xxx line and 4xxx line have a audio chip (usually Realtek I believe) for outputting digital audio over HDMI (only)".

So maybe you're missing a driver or something is wrong with the receiving side of the HDMI cable and it isn't set to accept sound. The sound should come out of your TV or whatever you connect it to. So "no sound at all on my computer" sounds quite OK at the moment.