The good news is that Romanian archival records are, after many many years of being purposely made hard to research, finally open and available to the general public.
The bad news is that they're only usually open to the public after 100 years, due to privacy laws. And the awesome archivist/historian who started loosening the rules to make it easier for genealogists and researchers was fired by the newer and more left-wing Romanian government group that came into power about two years ago. So while the "open laws" are still there, you may want to act fast in case the now-liberalized laws get tightened again.
Practically nothing in the country has been microfilmed, not even by the intrepid genealogists in the LDS Church (the Mormons), who run FamilySearch.org, because the government was so closed off for so long, even after the nominal fall of communism. I hope this will change soon, but who knows…
So, you'll probably have to go to the local archives to do your research, or find someone in the country to go for you.
Romania has 41 little local branch archives, all over the country. They tend to correspond to modern province borders, but not exactly. For example, Baia Mare's archives in the northwestern part of Romania also carry some books for towns that were part of the old Maramaros megye (county), Hungary, but which are now in sub-Carpathian Ukraine. So it's kind of hard to figure out what is where, due to border changes. You might want to call ahead and talk to an archivist.
If the books are more than 100 years old (i.e. pre-1914), they should in theory be available for you to research in person. In practice, however, a lot of books from 1906-1914 have not been made available yet, for whatever reason.
You can look at five books per day per person, so if you bring a spouse or friend, you guys can look at ten books per day together. You may need to give the names of the books you want at least 24 hours ahead of time, though, so factor that into your planning for any trips.
The great part is that you can digitally photograph every page of the books, if you bring your own camera! You can also Xerox pages, but they charge a small fee per page.
The only restriction is that you cannot publish the actual images on the Internet or in a book. This is pretty stupid, because it's not like they own the copyright on 100+-year-old government records, but whatever.
Note that second copies of the vital records books were kept in local town halls, and some of them may still be there to this day. If a book is in the town hall, it's up to the local government people in the building whether or not you can see the pages, whether or not you can photograph the pages (or just transcribe with a pencil and paper), and what cut-off year they will let you inspect. That is, you may (unofficially) be able to see information from much later in the twentieth century than the local archives branches will allow.