Boredom is part of the transition when you make this kind of change. They whine and carry on, and so you set limits -- like if you need to whine and complain, it needs to be done out of earshot of Mom. Have a (family) complaint book where they can write down how they feel when they are frustrated. Or maybe a once-weekly "complaint hour" where they can voice all their complaints at once and then they have to be done until the next "hearing." I refuse to listen to complaining/whining but they can tell me how they feel in a respectful tone of voice and I will listen to that.
My kids come back from their dad's house telling me they are bored. We have no TV, no video games, only very minimal Internet use and not for my girls (8 and 10). At their dad's there are a ton of neighborhood kids and they eat junk and convenience food. They go there for two 4-day stretches a month.
I just don't engage with their drama. If they complain, I reflect it back "I hear that you're bored; you wish we had TV here; you wish you had more kids around; you wish I was less strict about (fill in the blank)" and then I affirm my strong belief that they will figure out something to do. "Boredom is an opportunity; you will find something to do." It works -- nothing is more boring than lying on your back and looking up at the ceiling. Eventually they decide it's stupid and get up and do something else.
My standard line, which they all know well enough to say for me: "If you're bored, lie on your back, on your bed or the floor or the grass, and look up at the sky or the ceiling, and clear your mind. Eventually something will come to you and you'll figure out something to do."
I would also begin regularly culling (with them) their toys and things, if you feel they have too many. (Unless you think you can get away with culling the stuff they never use/notice/remember first.) Either donate a box to charity every 3 months (or month or some other timeframe), or make a rule that when something comes in something else has to go out, or do a twice-yearly purge before birthdays/Christmas, or whatever. Some schedule you and your kids can remember and stick to. If it's too hard to donate stuff yet, put it in a "holding bin" in storage where you can find it easily but the kids can't...if they don't ask for any of that stuff back in 3 months, it goes.
That sort of thing helped our family a lot in the early days.
If your kids need "reasons," make it about clutter, giving to those who are less fortunate (and there are so many), etc. They might not dig your personal simplicity agenda, but a family commitment to a clutter-free house and generous giving is harder to argue with.
Is your spouse on board? That makes a really big difference. It can be done either way, but if the two of you are on the same page first, you can both model things well for the kids.