Since last year, the Romanian Archives have simplified their procedure. Previously, non-Romanians had to apply to Bucharest for permission to research (a separate permission is needed for each archive).
The new procedure is that you can just show up and research on the spot (you show your ID and they issue you a research card). This is what I was able to do in Maramures, Hargita and Suceava. In Satu Mare, however, the director still follows the old system; it took about two months to receive the permission, and I had to make a second trip.
I am an American citizen, though I live in Budapest, Hungary.