Hmm... Part of me wants to push my organization (at least the mercy ships, which do a lot of medical and dental stuff), but there are lots of organizations out there. I'd look over some of them, write to them, get info, and pray about them. I'd say go in a christian organization so you will not be barred from sharing your faith. That much at least is important...
Actually, Mercy ships have split off from YWAM, but that's primarily to focus on the medical side and not so much the schools. You may want to contact them. I have a friend who was on a YWAM boat. He said it is strange being with people on the boats because they get addicted to sea living, but he fell in love with what he did and he's training to be a dental hygienist and will likely do more of that work later on.
http://www.mercyships.org/About/AboutMain.cfm
That's a good one, but don't just look at one organization... Check your denomination If you're a catholic, there's loads of stuff I would think, but most protestant denominations have mission careers if you can live with a paycheck rather than volunteer work... They may require a theology education, though... It depends on the denomination. Check out organizations that are Christian but not tied to any denomination, though.