  We prove that all discrete isothermic nets with a family of planar or
spherical lines of curvature can be obtained from special discrete holomorphic
maps via lifted-folding. This novel approach is a generalization and
discretization of a classical method to create planar curvature lines on smooth
surfaces. In particular, this technique provides an efficient way to construct
discrete isothermic topological tori composed of fundamental pieces from
discrete periodic holomorphic maps.
