A new model for circular cylindrical Kirchhoff–Love shells of flexoelectric–elastic materials with the centrosymmetric cubic symmetry is developed by considering both microstructure and flexoelectric effects. The couple stress theory is used to describe microstructure effects, and a curvature-based flexoelectricity theory is applied to account for flexoelectric effects. The governing equations and boundary conditions are simultaneously derived through a variational formulation based on Hamilton’s principle. The newly developed shell model recovers the model for Kirchhoff plates of cubic flexoelectric–elastic materials as a special case when the shell radius tends to infinity. To illustrate the new shell model, static bending, free vibration, and forced vibration problems of a simply supported axisymmetric circular cylindrical shell are analytically solved by directly applying the model. Numerical results reveal that the microstructure and flexoelectric effects lead to enhanced extensional and bending stiffnesses of the shell. In addition, the first and second natural frequencies of the shell predicted by the new model are found to be higher than those predicted by the classical elasticity-based model, but the difference is diminishing with the increase in the shell thickness. Furthermore, the results for both the mechanically and electrically forced vibrations given by the current model show that the deflection amplitude and the electric potential distribution in the shell are both frequency-dependent and can be tailored by controlling the excitation frequency. These findings indicate that a flexoelectric–elastic shell of a centrosymmetric cubic material can be used as a sensor for energy harvesting by converting mechanical energy to electricity or as an actuator by converting electrical energy to mechanical work.