Does it not occur to you that the most "elementary" proof might actually be geometric in nature? Or are we using Conservapedian definitions here?

This shouldn't be too tricky. Since the leaves of the foliation are regular, the foliation will induce a submersion from \R^3→X for some surface X (namely, the space of leaves of the foliation). π_1(X) and π_2(X) are both trivial. So, X is diffeomorphic to \R^2. From here, it's a matter of classifying submersions over the plane with 1-dimensional leaves with a given affine structure. If we can find a surface embedded in R^3 that intersects each leaf exactly once, then this can be used to give the submersion a line bundle structure which means it's trivial.

It looks like the paper you want to read is: Manifolds with Many Complex Structures by Dominic Joyce, in Quart. J. Math. Oxford, vol. 46 (1995), 169-184. It's available here: eprints.maths.ox.ac.uk/61/1/complex.pdf In it, he constructs examples of manifolds with a prescribed number of complex structures with the relations (and subsequent restrictions) as you desire.

Thanks for the citation -- I didn't know about this book, to my embarassment. I like the almost Zen-like redundancy of the tile.

Yes, I meant the edges to be line segments -- I have modified the question accordingly.