You can construct a translation surface from unfolding the triangle with angles Pi/12, Pi/3, and 7Pi/12. Alternatively, you can construct this surface by taking the regular 12-gon, and gluing equilateral triangles to each of the edges, and then identifying the triples of triangles that differ by a translation. The result is a surface of genus 4 and admits (infinitely) many decompositions into parallel cylinders. Some of these are pictured below:

If w is e^(i*Pi/6) is the 12th root of unity, then the cylinders are parallel to the vector 3+3w-w^3 in the complex plane.

Available for download as [ps] [pdf]

If w is e^(i*Pi/6) is the 12th root of unity, then the cylinders are parallel to the vector 4+5w-2w^3 in the complex plane.

Available for download as [ps] [pdf]

If w is e^(i*Pi/6) is the 12th root of unity, then the cylinders are parallel to the vector 5+6w-2w^3 in the complex plane.

Available for download as [ps] [pdf]

If w is e^(i*Pi/6) is the 12th root of unity, then the cylinders are parallel to the vector 6+5w-2w^3 in the complex plane.

Available for download as [ps] [pdf]

I decided to share these pictures, because I find them aesthetically pleasing. If for some reason you would like to create more you can take a look the software I used to make it: [tar.gz] (Though, the code is probably not very readable)