"Smooth surfaces of constant mean curvature (cmc) and minimal surfaces are well-known examples of isothermic surfaces. A discretization of minimal surfaces in terms of discrete S-isothermic surfaces was introduced by Bobenko, Hoffmann, and Springborn in 2006. These surfaces can be constructed from spherical orthogonal circle patterns, which serve as their Gauss maps. The orthogonal circle patterns themselves can be constructed from boundary data by minimizing a corresponding functional. Generalizing from minimal surfaces to cmc surfaces, the Gauss maps generalize to orthogonal ring patterns consisting of pairs of concentric circles such that neighboring rings intersect orthogonally. We present a method for constructing discrete analogues of given smooth cmc surfaces and see examples in R^3 and in Lorentz-Minkowski space R^{2, 1}. The image on the workshop website shows a doubly periodic S-isothermic cmc surface with hexagonal lattice symmetries in R^3. This talk is based on joint work with Alexander Bobenko and Tim Hoffmann. "