Hang three or four of your largest-sized plastic containers on the wall and label them. The nails will go easily through the plastic, and the containers and lids are so light that you won't need any anchors.
Decorate the plastic containers so that you have an organizational system in place.
Fill the containers on the walls with the appropriate lids. This works great if you have young children because they can help decorate, and it is a great way to involve them in kitchen clean-up because putting the right lids in the right containers can be a sort of game. Baskets work well for this same purpose, too.
Measure the length and width of the bottom of a dish-drying rack.
Cut heavy-duty cardboard according to the measurements of the rack, and fit it snugly into the bottom of the rack.
Measure the height of the rack.
Cut several more pieces of cardboard, measuring to the length of the rack, and turn it sideways so that it sits even with the top of the rack.
Fit your lids to the rack by placing some in and adjusting the spacing of the cardboard pieces. When you have a configuration that suits your needs, glue the pieces of cardboard to the bottom piece lining the rack. Decorative touches will embellish this and make it a perfect fit for your style.