Great idea!
I was trying to think of how I could "attach" the fabric to the door (was thinking some kind of adhesive, but afraid it eventually might bubble/peel). If I can find some bamboo sticks (or just buy some blinds and take them apart), I could probably do the same thing with those, and just glue them on as well. Then, if one should happen to lift or come off, it's easy to fix by re-attaching that one part, whereas the fabric would be harder to fix.
So many options ...