This is a common problem, but you can conquer it. This is a problem of not minding mom, rather than of being slobs.
Take a Sat AM to get each room cleaned up. Organize the toys: legos in a plastic tote, stuffed animals in toybox, etc. Get rid of unwanted or outgrown toys. If it is something special, a gift from someone or something they were really attached to, but are too old now and you want to keep it, put these in a plastic bag and box it up and put it in the attic, be sure to label the outside of the box. Clear out- tell each child they must get rid of 20 things (or whatever number) and put these in garage sale or goodwill.
Then, explain to you children that they are no longer allowed to destroy their rooms. After each play time, they must pick up before going on to the next thing. This is going to take work on your part too. If you need to take a child to an activity, you are going to have to give ample warning to get ready to go. It is going to take work for you to watch them, see who is leaving the trash, crumbs, junk laying around. If no one will admit to it, punish them all equally.
Also, before bathtime at night, all toys picked up and put in their designated spots. Then, bath, brush teeth, read and go to bed.
Now, how to get the kids to mind? Easy. For the first offense, stand in the corner for 1/2 hour, then when time is up, say "OK you can get out now as long as you pick up your toys". Second offense, 1 hour, third offense 1 1/2 hr. It probably won't take more than this.
You can make them sit on a chair facing the wall, or stay in bed for that amount of time, but it should be away from the other kids and no TV allowed.
NEVER yell out of control; just keep a firm, calm, matter-of-fact voice.
Punishment is the best way to make kids mind. The punishment must DETER the crime.
Make cleanup an everyday thing for your kids. Have them help with clearing the table, doing the dishes, sweeping, cleaning the bathroom sink, etc.
I have watched my daughter train her 2 1/2 yr old to pick up after himself; if he can, your kids can. She has put most of his toys in his closet. He can't open the door. When he wants to play with something else, she tells him "you know the rule, pick up what you have out, then you can get something else". And limit what they get out. If they are playing with cars, don't let them get all the stuffed animals out too. If they are playing house and have the dishes out, don't let them get the Legos out with it.
Hope this helps!