It wasn't my idea, but it works.
Since the Deputy Headmistress asked...
While I was in the States last year, we spent a few days in the home of a pastor. He and his wife had eight children, ranging from early-elementary to college age. A few times while we were there, the mother of the home announced "everyone spend 15 minutes cleaning your room!" Without further instruction, and within a very short time, the entire house was tidy. "Your room" did not mean "your bedroom" (none of the children had rooms of their own), but rather "the room assigned to you."
Each child was responsible for a room in the house. Every public room--kitchen, bathrooms, family room, living room--was covered, as well as hallways, entry-ways, and the steps. Older children had tougher areas like the kitchen, while younger children were assigned simpler areas to clean. I was so impressed with her system, that I actually wrote it down, so I wouldn't forget about it before I got back home to Poland.
I have far fewer children capable of work, but I divided my house into four "sectors" (room by room didn't work for me), and assigned one to each of my three older children, and one to myself. Those sectors are their spaces to clean and keep tidy. We spend 30 minutes per day on "sector time," so I effectively have 2 man-hours of cleaning done each day. They keep the same sector for a long, long time (many months), so they learn to cover daily cleaning, weekly cleaning, and the deeper cleaning that only occurs a few times per year for that sector.
Any time I need the house to look better than it does, I can announce, like the mother who gave me the idea, "Everyone spend 15 minutes on your sector!" and within a very short time, the house looks quite tidy. No one is allowed to ask someone else to clean their sector during an official "cleaning" time. If someone else made a mess in your area, you just have to clean it up. However, you may also "police" your area and ask anyone (even a parent!) who has left a mess in your sector to clean it up at other times.
Now, I have to admit that this does absolutely nothing about untidy bedrooms, but it does keep the public areas of the house in reasonable order, and I always know that within a short amount of time, everything can be improved. No one is getting in anyone else's way, and there is no arguing about who should be doing what job.
We've had this system running for over six months, and it's still functioning, which is probably some kind of record. My bathrooms are much cleaner, my trash cans don't overflow, and the dust actually doesn't have time to build up.