Inspired by the helix-shaped microstructures found in many collagenous tissues, a class of three-dimensional (3D) soft network materials that incorporate similar helical microstructures into periodic 3D lattices was reported recently. Owing to their high stretchability, high air permeability, defect-insensitive behavior, and capabilities of reproducing anisotropic J-shaped stress–strain curves of real biological tissues (e.g., heart muscles), these 3D soft network materials hold great promise for applications in tissue engineering and bio-integrated devices. Rapid design optimization of such soft network materials in practical applications requires a relevant mechanics model to serve as the theoretical basis. This paper introduces a nonlinear micromechanics model of soft 3D network materials with cubic and octahedral lattice topologies, grounded on the development of finite-deformation beam theory for the 3D helical microstructure (i.e., the building-block structure of 3D network materials). As verified by finite element analysis (FEA) and experimental measurements, the developed model can well predict the anisotropic J-shaped stress–strain curves and deformed configurations under large levels of uniaxial stretching. The theoretical model allows a clear understanding of different roles of microstructure parameters on the J-shaped stress–strain curve (that is characterized by the critical strain of mode transition, as well as the stress and the tangent modulus at the critical strain). Furthermore, we demonstrate the utility of the theoretical model in the design optimization of 3D soft network materials to reproduce the target isotropic/anisotropic stress–strain curves of real biological tissues.