For Copenhagen, you can often get the 4* Copenhagen Marriott on Priceline for $95 US+tax/night or so. (See www.biddingfortravel.com) No guarantees it will be the Marriott, of course, although no other Copenhagen hotel has ever come up. If you bid for only a 4* hotel, I doubt you will go wrong in any event. If you want anything much cheaper, I fear you won't have much luck at any star level. Ibis hotels are about the cheapest for conventional booking - same in Stockholm.

(I don't know how big a concern price is in your case, but I'm also planning a multi-city Scandinavian trip where price is a concern, and it's been a real challenge to find good deals!)