It won't be easy as I think there's a fair amount of bureaucracy involved. First thing to do is to contact the Romanian embassy in your own country and ask them what papers you will need to apply for a licence to get married in Romania. Secondly, try to find some travel agencies in your country that specialise in trips to Transylvania - it could be that some of them have done this before and if not they might be willing to help - if only because it could generate some good publicity for them. One agency that has done this once before is International Tours and Events - they helped someone get married during a Dracula Tour. Their website is: http://toursandevents.com/2005Dractour.htm
Even if that is not exactly what you were after, they may be able to give you some good advice.
If you want a church wedding as well as a civic ceremony, you'll probably also need to contact a church in Romania and get a letter from your local minister.
Probably the best place to aim for in Transylvania is somewhere like Bran castle it's reputed to be 'Dracula's castle', though there are a few placed in Romania that make this claim.
I found one set of contact details for someone in Romania who may be able to help organise a wedding here: http://www.afwpi.com/europe/romania.html
This site is more general, but you may get more help in their forums: http://www.weddings-abroad-guide.com/
I think you'll need a lot of patience and a lot of savings, so good luck!