Simple shearing flow of a ‘‘dry,’’ perfectly ordered, three‐dimensional foam composed of planar films is considered. The undeformed spatially periodic cell structure is formed by regular tetrakaidecahedra, which have six square surfaces and eight regular hexagonal surfaces. The elastic–plastic response of the foam is modeled by assuming that all surfaces remain planar and that the angle between connected surfaces does not change during elastic deformation. An explicit expression for the stress tensor that is valid up to the elastic limit is determined. Past the elastic limit, the foam structure and macroscopic stress are piecewise continuous functions of strain. Discontinuities in structure and stress are associated with topological (T1) changes in the film network structure that occur when the area of an individual film vanishes. These T1 changes, which reduce surface energy and result in the switching of cell neighbors, are essential mechanisms for yield behavior in foam flow. The foam structure is determined for all values of shear strain by choosing initial cell orientations that lead to periodic behavior with strain. The shear stress evaluated from a strain energy method differs from that obtained by volume averaging the local surface tension forces; this inconsistency arises because a foam with planar films cannot satisfy the equilibrium requirement that three films meet at equal angles of 120°.