I use brown paper - I'm not keen on scents going through the baby clothes.

For drawers where I'm keeping other stuff (not clothes; eg. bottles, lotions etc) that I don't want to roll around, I put a layer of brown paper covered by a layer of this rubber stuff from Big W - it comes on a roll in various colours, specifically for putting places to stop things rolling. It's a mesh, you can cut it with scissors.
What a GREAT idea - I know that mesh stuff your talking about too