As with anywhere in eastern europe, the turning up at the station and being harrassed by the old ladies with rooms is the best way to get a good cheap place to stay. Try to get hold of a map first though, because they don't all speak english and you don't want to be too far from the old town unless you want to rely on busses.
Seriously they are there and you will find a room, if you don't like it you leave the next day and find another one, there's loads all over Dubrovnik.