D.W.
I'd either recommend getting a nice box (the size of a shoe box) for each kind of card (birthday, holiday, sympathy, etc) or using plastic bags.
Some office supply stores (not your main stream ones) carry plastic zipper bags in a variety of sizes). If you organize them by type, date, etc. you can put them in a bag, label the bag, and then file in a bankers box, etc.
It also helps protect the cards/letters from time and normal elements.
Good luck with your move.