CStanford
If you read some of the replies in too granular a fashion you could come to the ridiculous conclusion that drawers never stick with changes in humidity, etc. and that nothing matters and practically anything goes. This is clearly silly on its face and undoes centuries of standard trade practice in fine cabinetmaking.

Unless your goal is to make drawers that swim in their openings to accommodate potential movement, I'd use quartersawn material around 3/8" to 1/2" thick maximum for sides and backs. And especially so if you already have the material on hand and available for your use.