Texture-based magnonics focuses on the utilization of spin waves in magnetization textures to process information. Using micromagnetic simulations, we study how (i) the dynamic magnetic susceptibility, (ii) dispersion relations, and (iii) the equilibrium magnetic configurations in periodic magnetization textures in ultrathin ferromagnetic films in remanence depend on the values of the Dzyaloshinskii–Moriya interaction and the perpendicular magnetocrystalline anisotropy. We observe that for large Dzyaloshinskii–Moriya interaction values, spin spirals with periods of tens of nanometers are the preferred state; for small Dzyaloshinskii–Moriya interaction and large anisotropies, stripe domain patterns with over a thousand times larger period are preferable. We observe and explain the selectivity of the excitation of resonant modes by a linearly polarized microwave field. We study the propagation of spin waves along and perpendicular to the direction of the periodicity. For propagation along the direction of the periodicity, we observe a band gap that closes and reopens, which is accompanied by a swap in the order of the bands. For waves propagating in the perpendicular direction, some modes can serve as unidirectional channels for spin waves. Overall, our findings are promising in sensing and signal processing applications and explain the fundamental properties of periodic magnetization textures.