I can't seem to get a straight answer with Google and searching these forums results in endless comprehensive bug lists. I'm on the 360 so no fancy console tricks .

I just need something. A simple dresser would do. Help?

Well, are there any bed that are not listed as "owned"?
Typically safe storage is near those.

Unless it is marked owned, all furniture in the game is safe to store stuff in. None of the furniture is set to respawn. Only Barrels and Crates and Chests are set to, and not all of them are, but just avoid them in places you don't own.

Note, not all places that are not marked as owned are safe with regards to item ownership. I have heard that folks who use the Thalmor HQ in Solitude have their items marked as stolen. So, test with a minor item.