The band structure of spin waves in chiral magnonic superlattices is theoretically studied. The proposed metamaterials involve a thin ferromagnetic layer in contact with a one-dimensional array of heavy-metal stripes, causing a periodic interfacial Dzyaloshinskii-Moriya interaction. Two superlattice configurations are examined in this research, with attention given to either heavy-metal stripe extraction from the supercell, which contains several stripes, or modification of the central stripe width. For the first class, dispersionless modes are found within the band gaps associated with the regular chiral magnonic crystal. A fine structure is also obtained for the low-frequency flat bands, where a set of subbands shows the typical quantization of laterally confined waves. In the second class of superlattices, defect modes are observed, which have properties similar to those seen in nonchiral magnonic crystals with defects. Nonetheless, additional low-frequency flat modes with a notable localization in the defect region are predicted, which emerge at a frequency even lower than the lowest band of the defect-free chiral magnonic crystal. Micromagnetic simulations agree with the predicted behavior from the plane-wave method calculations, reinforcing the theoretical modeling of the chiral superlattices. These results are relevant for developing magnon-based devices relying on nonreciprocity and channeling of the spin waves.