I have a lot of little people in my house, and for now they are all young enough to adore me. I get cards and pictures like you wouldn't believe. I don't think children should feel like they are the centre of the world and all their art should be cherished as if drawn by Picasso himself, but they did make something they're proud of and probably shouldn't see it sitting in the recycling the next morning.
I'm pretty honest with my kids. Age appropriate honesty of course. I don't like recycling their art without telling them that's what is going to happen, so if they're feeling attached to what they've made I take a picture with a digital camera and then recycle it. They feel happy that their paper is going off to a plant to make more paper, and also feel like all their work wasn't for naught.
This was a sign my son wanted me to take a pic of and post to facebook so he could "get work". Is this one saved on the external hard drive? You betcha. The print says he will work for one week, for $5.00 - but that's only for people 45 and under, for "old" people it's free!
I usually wait until our art table is crammed with paper (every four months or so), take 30-40 minutes and sort out all the artwork and take digital pictures of all the "special" pieces. Computers have a lot of digital space these days, and if yours doesn't there are external hard drives and online storage like iCloud. Your kids will get to "keep" their art, and you get a tidy art corner again!