Leave Stuff in Drawers
If you want to save boxes and the trouble of moving items from a drawer to a box, then simply take the drawers out of chests and desks. As long as the drawers are full and don't contain anything breakable, you can stretch tape across the top of the drawer to hold the components in place. Be sure to use masking tape -- it won't take the paint off of the furniture. That said, don't leave the masking tape on the furniture for too long. Take it off as soon as you move the drawers into your new place.