The concrete idea is good, but then there is no place for your heat ducts and pipes. Can you dig down a bit to give you some room to run pipes and whatnot (and perhaps access them in the future). This would also give you room to lay rigid insulation (expanded polystyrene) and a vapour barrier on the ground and against the foundation, which would make for a much warmer floor.
I would eliminate the existing structure due to rot, and go with a proper joist setup. Professionals aren't needed, just access to the National Building Code and the urge to learn and read. A book I highly recommend that you buy is the Canadian Wood Frame Construction
book which talk simplifies the tables in the NBC for the average homeowner.
I have no idea what the size of the room is, but the above book would easily tell you the size of floor joists you need (i'm guessing 2x6). They would either rest on top of the foundation (depending on the height of the adjacent floor and if you want a step in your floor) or attached the joists to a ledger board which is bolted to the side of the foundation (there are screws made specifically for anchoring into concrete), or a combo of both.