I'm a Vietnamese national currently residing in the USA. I applied for a DC visa to study abroad in Germany (with the German Consulate in San Francisco). Under this visa, can I fly from the US into Rome and visit relatives in France and Italy for a few weeks before heading into Germany?

Thank you, and I look forward to your answer!

Yes. You can travel in the Schengen area during 3 months before heading to Germany.

As far as I know, the Schengen visa is good for the whole shengen zone countries.