The current work investigates the wave dispersion characteristics of two-dimensional structures, accounting for the effect of their higher order inner material kinematics. For the computation of the nonlinear dispersion diagram, a perturbation approach appropriate for the incorporation of nonlinear effects on the linear band structure attributes is employed. The method is used to compute the dispersion characteristics of architectured periodic materials, structured with hexagonal, re-entrant hexagonal, as well as square and triangular-shaped unit-cells. The corrected nonlinear dispersion characteristics suggest that the incorporation of the higher order kinematics induced corrections, entail a wave amplitude and wavenumber dependent mechanical response. Furthermore, the numerical simulations demonstrate that nonlinear effects primarily arise for the lowest rather than for the higher eigenmodes. What is more, it is shown that the highest magnitude corrections are expected for the lowest shear mode in the low frequency region, while for a given wave amplitude, the unit-cell design plays a significant role in the magnitude of the obtained nonlinear correction.