Spin waves excited in periodically modulated magnetic nanomaterials, known as magnonic crystals, exhibit characteristic band structures. These bands can be tuned by material engineering and have been attractive for potential spin-based applications. When periodic nanomaterials with handedness are introduced, spin waves inherit the chiral feature in their behavior and manifest an exciting range of novel physics, including asymmetric and unidirectional propagation, low-frequency magnonic flat bands, and indirect band gaps. This study investigates the properties of these chiral magnonic excitations. The analysis is performed in ferromagnetic films patterned with nanowires of two different materials that produce periodically modulated perpendicular magnetic anisotropy and interfacial antisymmetric exchange (Dzyaloshinskii-Moriya interaction). The low-frequency flat modes are studied using a magnonic localization diagram that distinguishes the spatial confinement degree in zones with and without antisymmetric exchange. An analytical expression is derived for the transition region in the localization diagram that outlines the zones where magnonic confinement occurs. The findings reveal the presence of flat modes with nonreciprocal magnetization oscillation amplitudes between waves with opposite propagation directions when the spin-wave localization occurs in regions with Dzyaloshinskii-Moriya interaction. Conversely, reciprocal oscillation amplitudes are observed when modes localize in the nanowires with perpendicular anisotropy. Micromagnetic simulations demonstrate the amplitude asymmetry of the flat modes, yielding perfect agreement with the theoretical predictions. This paper provides a deeper understanding of the behavior of spin-wave modes in chiral magnonic crystals and establishes a method to control their associated magnonic bands for designing spin-wave-based nanodevices.