Constant mean curvature surfaces are æsthetically appealing geometric objects with applications in physics, differential geometry, and architecture. We present a simple discretization of constant mean curvature surfaces based on the classical observation that their Gauss maps are harmonic. Our construction is elementary---requiring only discrete Dirichlet energy minimization and a Poisson solve---yet it exactly mirrors the smooth theory. A discrete analog of conjugation produces discrete constant Gaussian curvature surfaces. Their unit offsets are discrete CMC surfaces with a conformal parameterization. Additionally, we introduce a novel Möbius-invariant discretization of the Dirichlet energy for sphere-valued maps on dual meshes that is derived from the discrete Willmore energy. It is more robust than standard formulations based on inverse cotangent weights. Our approach to the construction of constant mean curvature surfaces provides direct control over tangent planes along a boundary, if present, and naturally handles closed and periodic examples. We demonstrate the approach on a range of free-boundary, symmetric, and periodic CMC surfaces.