I have a couple of large tubs that I store toys in so I always have something to rotate in and out when he gets bored with the toys he's been playing with for a while. I also keep all the little pieces for stuff (like little people) in ziplock bags labeled with the toy the belong to. We are lucky enough to have an extra room that is our playroom. We took the doors off of that closet and put some shelves in there all the way across, and keep a lot of toys (and books he hasn't grown into yet) up there. That way he can see them and can ask for them if he wants them, but we're not always tripping over stuff. And if I take something down, I try to put something else back up. Of course we still have plenty of toys everywhere, but it helps some.
Also, I try to keep a few carefully selected toys in the living room like 1 good ride on toy (like a bouncy horse) and we have wood floors so I pick some good cars that will work well in here. My son has the 2 bottom shelves of the book shelf in the living room. The very bottom shelf has a nice basket where I keep the cars and stuff, and the other shelf can hold 2 or 3 other mid sized toys that don't have too many pieces. With 3 kids, I'd pick something like the cars that they can share, and then let them each pick one or two more that they can have in the living room, and they can switch these out. Maybe get some of those nice leather (or leather-like) ottomans with the removable tops to keep toys in. Maybe even each child can have their own. That way it stays looking nice, but they can have stuff to play with too.
In his bedroom I keep quiet toys for playing with after bath time like puzzles, books, and other quiet activities. Everything that plays music, lights up, has a million pieces, etc, is in the playroom. That way at least if any room in the house has to be chaotic, it's the playroom.
Whether you have a playroom or not, organization is key. Shelves, tubs, buckets - whatever. A tall shelf or two where you can put toys the kids can't reach is great too, so you can be in charge of when those come down.
The idea of putting the rings through the ziplock bag and hanging it in the closet is genius! I will definitely have to keep that one in mind too!