Metamaterials are mostly studied using the time-harmonic approach, where the wave propagation is spatially described. Recently, studies of media having electromagnetic properties that change in time have been given the attention of the scientific community, aiming to describe wave-matter interaction in both space and time. In the present paper, we use the space-harmonic method for the general description of wave propagation in time. Such a method can be used to effectively describe a "temporal multilayered" (or temporal multi-stepped) metamaterial by alternating the effective material parameters of the medium in time between two values. We obtain the exact analytical solution for the fields in two examples of temporally periodic metamaterials. Numerical simulations for the impedance-matched scenario are also carried out, showing an excellent agreement when compared to the exact analytical field. Our results also demonstrate the duality between the descriptions of the temporally periodic metamaterial and its spatial counterpart, spatially periodic metamaterials.