Dressers, chairs, night stands ... you've doubtless dusted, cleaned and waxed all of these items already. So what sets the bedroom apart from other rooms? The bed, of course! The bed frame counts as a piece of furniture, so don't forget to pay close attention to it as well. But by far one of the biggest cleaning issues in the bedroom is the mattress. The truth is, mattresses can get really musty. If you're just changing the sheets and pillowcases every few weeks, you're not getting the mattress itself clean. Not to get too disgusting, but while you sleep, you're shedding skin cells and hair and maybe even sweat on this highly-absorbent pad for however long you've had the mattress. And if you have small kids or pets...that mattress has probably seen some bad stuff.
There are two ways of going about this. Either way, you should flip your mattress every six months to promote even wear. But after taking off the sheets and flipping? The involved way to clean it is to vacuum it first, and then get out the carpet cleaner with the upholstery attachment, using stain remover on any spots. Make sure not to get it too wet and allow plenty of time for it to dry -- if you can, open windows for it to air-dry or use fans. If the mattress isn't stained but just a bit smelly, you can just sift baking soda over the top after vacuuming it. Then put the sheets back on. The next time you change them, vacuum off any baking soda residue. It'll have absorbed odors and moisture. Pillows are the second-worst-offenders -- they can usually be washed but you may need to take them to a Laundromat to use the big commercial washers and dryers.
Closets are often ignored during a bout of spring cleaning. Out of sight, out of mind? Remove everything from your closets and clean them as you would any other room. Since warm weather is on the way, go through your winter stuff and decide what should be packed away and what needs to leave the house one way or another (and don't let your good intentions die with bags and boxes sitting on your closet floor until fall). Then organize what's left in your closet. Extend this work to linen closets, coat closets and any other closets, while you're in the room that contains it. You will probably run across some "lost" items, and be able to fit more stuff into your closets!