OK, so I don't think I've ever seen a topic like this started in my time here.
I'm pretty sensitive to almost everything that is in the air, and my birds are in my room, so I really have to keep the mess down. Here are my tips. I'd be really interested to hear what others do:
1) This is the best - I bought a large plastic sheet from Crappy Tire (hardware store) - it comes in these enormous rolls, and you buy it by the yard. You have to get the staff to cut you a piece. It's just a clear plastic sheet. I bought a really long piece - about 9 feet, and hung it on the wall, about six feet up, using velcro with adhesive on one side, that I bought at a fabric store. I didn't want to hang it with nails, just in case one ever fell out - I wouldn't want it to end up in the cage. Anyway, six feet of this sheet are up on the wall, behind the cage. At the floor, I just fold the sheet so that it goes along the floor for three feet, underneath the cage. I never have poop or bits of celery or broccoli on the wall or carpet. Every week, I remove the cage and spray down the plastic sheet with a water bottle. I wipe away the feathers, seed husks and dust with paper towel.
Edit: There are various grades of plastic sheets - buy the thinnest, cheapest one you can. This will mean it is not as heavy, and that you are more likely to be able to just hang it using a hammer and nail, rather than having to use anything stronger, such as adhesive velcro. I just hung a 9-foot by 4.25-foot sheet with two litltte nails, and very minimal damage to my wall. I folded one edge over about an inch to make a double thickness layer, and drove a nail into the edge, through the double layer (I was afraid that putting a nail through a single layer of the sheet woudl cause it to tear). I hung the sheet with nails high enough that the birds cannot reach them. If they were to find them, I could cover the nails by hanging a picture over them.
2) When I begin cleaning, before I even touch the cage, I thoroughly spray the whole thing down with my handy dandy water bottle. This weighs down all the dust, feathers, and seed husks so that they don't end up in my lungs.
3) I use a Swiffer to quickly rub down the walls closest to the birds, and the blinds on the window beside them. It takes all of two minutes, but you should see all the dust that comes off each week, that would otherwise end up - you guessed it - in my lungs!
My husband is allergic to cats, and I am worried that he will become allergic to the birds. One day, hopefully we'll have a bigger place and the birds will have their own room, but in the meantime, I think it's really important to keep the dust and mess levels to a minumum, for sanitation as well as health.