"Smooth minimal surfaces foliated by a family of spherical curvature lines were classified by Dobriner and Wente. Renewed interest in these surfaces stems from the finding that they provide special solutions to free boundary and capillary problems. In this talk, we shall generate discrete minimal surfaces with one family of spherical parameter lines. The construction exploits the fact that minimal surfaces belong to the integrable class of isothermic surfaces and is based on the concept of lifted-folding. The latter is a recently developed method that allows discrete isothermic surfaces with spherical curvature lines to be built from specific holomorphic maps."