Getting a dvi to hdmi adapter would mean having no audio over the hdmi cable. Meaning you would have to run audio either through a separate receiver or run it through aux in on the TV. Aux in isn't a supported by all televisions, hence why I recommend buying a card with HDMI and audio pass through if you intend to use the computer as a media center. Saves you a lot of headaches.
If you just want the picture without sound, iam2thecrowe is right, you can just buy a dvi -> HDMI cable. They retail for about $4 for 6 feet on amazon.