Ask a chef about what s/he does with his/her pots.... a good one will say "It's not what you do with your pot, it's where you put your lid!"
This is, naturally, a wise model to live by. Here's a pot lid solution from salvaged towel racks.
Step 1: Materials
I found these really cheap (as in quality) wall mount racks ready for incineration/landfill space. These are perfect!
1: Some form of towel rack
2: Mounts for said towel racks
3: Velcro or screws
Step 2: Hack to Size
My racks wouldn't fit on my cabinets - so I took m hacksaw to them to remove a bit of material. These racks are made of two interlocking channels that can extend outward. Measure twice, cut once... or in my case - guesstimate, cut twice, fit on there and cut one more time.
Step 3: Attach
I attached some Velcro to the back of my fasteners - I can't destroy this rented apartment too much.
Here's something I learned awhile back in my engineering courses.....
When you apply a shear load, the stress will try to peel away from the top surface downward. This is why hooks and such have their screw mounts ABOVE the load rather than below or at the load point. So don't be tempted to apply a small amount of Velcro - apply a bit above the load point.
OR, screw into your cabinets - I would have if I could :)
Step 4: Alternate Location
I like my lids - they look good :) And I'm not afraid to show them off (I don't have enough inner cabinet space anyway :P). So here's another location to show off your rack. This time, I used screws and anchors directly into drywall.