Cheap accomodation is often linked to non-furnished appartment.
So in case you find one that you like you can always rent furniture and buy it afterwards if you feel like staying longer in Luxembourg.
Check www.interiorrent.com for info.
if you're looking for something cheap you need time. Best/ cheapest solution is to rent a house with a few people. If you search long enough or if you are just lucky you may find something "cheap" (means about 10-12 € per sqare meter and not equiped at all). A quite cheap place is Bonnevoie with is 10 min to the city center (by feet). Some say it is the worst part of the town but if you don't mind some junkies running around and the few planes passing by you will feel quite good.