Everyone, STOP!
DVI and HDMI are electronically compatable, but only HDMI can carry audio. DVI CAN NOT CARRY AUDIO SIGNALS, PERIOD!
ATI cards and some NVIDIA cards feature an HDMI IN that will take a HDMI audio signal (either directly with ATI, or with an internal SPDIF cable that connects to the motherboard with NVIDIA). So theoretically, you can take a HDMI signal, decode the audio portion, and output the video to a moniter using DVI. The audio signal, however, will be handled internally and is not re-transmitted.
Also, there should be zero compatability issues using a HDMI->DVI/DVI->HDMI adaptor, as for video, they are the exact same signal, just a diffrent port. You can get some signal degregation due to the adaptor itself though...