I keep cards that have a personal, handwritten message that is sweet or unusual and toss the rest. Sometimes I'll put them in our photo album. (Especially if it is, say, a birthday card from my SIL, and I have a photo of me with SIL at my birthday. Then I'll put the card and the photo together.) I have a file in my file cabinet for "keepsake correspondence," which is where I put the handwritten ones that don't make it into the photo album.
For Christmas cards, I use up a few pages of our photo album and keep the photos that people send, and I cut out their signature and handwritten message (if there is one) from the card to caption the photo. I also use a page or two of photo album just for the cut-out messages that people have handwritten in cards, if they say anything memorable. Otherwise, toss, toss, toss!
And for the record, I'm not a scrapbooker, I just have a regular old sticky-pages photo album where I put photos and basic captions, and using the cards actually makes it easier because I don't have to worry about captioning.
Homeschooling mama to DD 3/28/06 , DS 2/27/10 , Belle the Orange Dog 03/11, and DD 10/03/2013.