Check with Autoeurope and the big brand rentals. I rented from Thrifty in Hungary and used the car in Romania with no problems in terms of fees or the right to take the car across a border.

I'm no expert, but I had problems when I wanted to pick up a car in one country and return it in another. I don't think there's a problem driving between the countries, so long as you will be returning the car to the same country where you picked it up, even though you'll be dropping it off in a different city.

If someone else has a different experience, I'm sure they'll let you know on this thread.

This spring we rented a car in Vienna for travel in Austria, SK, CZ, HU, and PO. They wouldn't permit Romania. I have this feeling that sometimes the firms allow "non-standard" country travel, sometimes not. I suspect the same firm could have different policies on different days, maybe even from different clerks. So, try various companies, and don't assume that what one company says one day will be the same as it says another day.