There's a few ways to do it (and the press release is surprisingly good and mentions most of them), but certainly one of the more obvious explanations is that the emitting object, is in a binary system with some other object, and that modulates the emission mechanism somehow.

I don't think either needs to be a black hole though. What I consider the most plausible candidate for emission is probably magnetars (i.e. neutron stars with extremely strong magnetic fields), and there's no reason the companion would need to be a black hole either, you just need a companion that gives a 16 day orbit and that's very easy to do with another star or neutron star.

Or one thing rotating.

You can imagine other cyclical processes, but I don't know any are realistic.

Maybe, but perhaps not in the way you think. A source that emits radiation only at certain angles (think a stellar disco ball) would only "flash" us infrequently as we moved relative to it.