There's a good selection of furniture shops around the harbour area. One of the nicest is Dalmadom, it's a department store and has a selection of kitchen, lounge, bedroom etc ranging from economical to good quality. There is a couple of shops in a centre near the Tomy supermarket, Mimo is good fop cheap furniture, check out the bathroom store next door!
There is another good department store, not obvious from outside but at the bend of the harbour (near old bus station) there is a restaurant with glass front and upstairs is the store. Near this one is a family run place which is good for beds.
If you are looking for real class, Italian design, checkout Perak Interijeri, near to the solitudo development.
I'm not sure where in Dubrovnik your place is but I'm going back out to furnish my apartment (Solitudo) in April. Hope this helps.