I was recently asked this when I was bragging about keeping my home neat using the Flylady program.
I don't get my kids to clean. I ask them to clean up after themselves. Right now, that's a challenge in and of itself. It's not their job to be my maids any more than it's my job to be theirs.
I ask them to put their own laundry away, clear their own plates from the table, and bring the laundry downstairs. They are to keep their own rooms clean, and clean up their own dirty clothes out of the bathroom when they're done with baths. They make their own beds and change their own sheets.
I occasionally ask them to take out the garbage, and they think loading the dishwasher and doing laundry is pretty cool. So I let them do that, and praise them to the skies when they do.
I remind them often that "housework done improperly still blesses the family" and, while I do teach the correct way to clean things, I also tell them that it doesn't have to be perfect. Just better.
And I never, ever use cleaning as a punishment. I want them to like cleaning! I use it as a reward! I also never nag them about any jobs more than putting their own things away. You know, teaching them to take care of their own messes is teaching responsibility. Ultimately, if everyone cleans up after themselves, there's not much left to do!
I talk to them about how much better the house (or their rooms) feel when it's clean. It's so much nicer to be able to walk around barefoot without stepping on Legos and getting hurt.
I let them see how hard I work. I talk to them about my routines and Flylady. "See? Flylady says to shine my sink - I guess I'd better go do that!"
And I help them declutter. Mercedes just went through her clothes and got rid of everything that she doesn't love or that is too small. Jared got rid of some toys he doesn't like any more because he's outgrown them. Clutter can't be organized, just gotten rid of.
Most importantly, I set the example. When I was messy, so were they. When I'm working at FLYing, so do they, and without any prodding from me.
So start with making them responsible for their OWN things and let them grow into the routines. And remember - they aren't interested in cleaning the house, you are. Show them how desirable a clean, neat house is, and they'll follow through on that.