There does indeed. You can form a discrete minimal version of the Diamond (Schwarz D surface) by rearranging the faces of a rhombic dodecahedron.
I posted here an example of subdividing and optimising this to make a discrete minimal surface:
You could also just apply Catmull-Clark subdivision to the simple quad mesh - this won’t give an exactly discrete minimal surface with planar faces like the Koebe polyhedron method linked above, and the faces won’t be planar, but it’s a fairly close approximation.