From polymer gels to cytoskeletal structures, random networks of elastic material are commonly found in both materials science and biology. We present a three-dimensional micromechanical model of these networks and identify a “bending-to-stretching” transition. We characterize this transition in terms of concentration scaling laws, the stored elastic energy, and affinity measurements. Understanding the relationship between microscopic geometry and macroscopic mechanics will elucidate, for example, the mechanical properties of polymer gel networks or the role of semiflexible network mechanics in cells.