Hexasuccessagon!
I was bored at work today and had some free time, so I finished up the model. Printed out three of them tonight (didn't bother with a test) and the resulting flexagon is a little loose but definitely functional! It's a pretty nifty print, probably the coolest thing I've designed from scratch.
Edit: Here's the Thingiverse page, with a picture.
Edit: Here's the Thingiverse page, with a picture.