Good Question: Natural Fiber Containers for IKEA's Billy Bookcase?

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Lee asks:

I have an entire wall full of Billy bookshelves in birch from IKEA. I'd like to make better use of the bookshelves by using low-sided boxes and magazine holders to contain some of the stuff I use at my desk, but I can not find much in a natural fiber that will fit on the 10 1/2" deep shelves without hanging over. I did see a series at West Elm that would work, but only the CD box and magazine holder are the right size. Do you have any ideas?

Here's the canvas storage series from West Elm:

Lee, your idea to use low-sided boxes is right on, but we're having a hard time finding ones that will fit on the relatively narrow BILLY shelves. One option from IKEA might be the STRIKT Box with Lid, which is made out of cotton dyed in bright colors: