The translation invariant analogue of the Costa surface was constructed by Michael Callahan, David Hoffman, and William Meeks in 1989. When divided by its translational symmetry, it has two planar ends and genus 2.
It is a bit harder to construct than the Costa surface, because one has to solve a 1-dimensional period problem. On the other hand, its embeddedness is easier to see, using the conjugate surface. The latter solves a curious free boundary value problem with a 12-gon in the interior and a planar end.
As for the Costa-Hoffman-Meeks surfaces, there are higher dihedral symmetry versions.
Mathematica Notebooks k=2 and general k
Mathematica CDF k=2 and general k
M. Callahan, D. Hoffman, W. Meeks: Embedded minimal surfaces with an infinite number of ends, Inventiones Mathematicae 96 (1989), 459-505.