A microstructural model for the rheology of large‐gas‐fraction foams and concentrated emulsions is developed. Large shear and extensional deformations of a two‐dimensional spatially‐periodic network consisting of monodisperse hexagonal cells are considered. The elastic response is determined by surface tension forces and the steric interaction of thin liquid films. Coalescence and disproportionation of Plateau borders result in the relative separation of cells and provide a basic mechanism for yielding and flow. The strain dependence of the macroscopic stresses and cell morphology is very sensitive to the initial cell orientation. The response is strain periodic for discrete values of the orientation angle; however, strain‐periodic orientations for simple shear and extension are mutually exclusive. The steady‐flow material functions are determined by averaging the instantaneous stress over the strain period. Three different physical interpretations of the yield stress are considered.