This work investigates how unit-cell topology controls flexural-wave dispersion and finite-array attenuation in periodic acoustic-black-hole (ABH) plates. Six configurations are compared under identical material constants, lattice parameter, power-law profile, and residual thickness: a uniform reference plate, two isolated ABH inclusions with circular and y-elongated elliptical shapes, a crossed-strip ABH network, and two cross-connected inclusion–strip networks. The plate is modeled using Kirchhoff–Love theory with spatially varying thickness, and dispersion curves are computed with a finite-difference Bloch (FD–Bloch) formulation along the sampled Γ–X–M–Γ path. The numerical dispersion results are checked against a Gaussian-enriched Ritz method and, for the uniform plate, against the analytical folded dispersion relation. To connect the infinite-periodic predictions with practical attenuation, finite arrays are also analyzed through transmission maps and full-aperture transmission curves. The results show that isolated circular and elliptical ABHs mainly distort dispersion branches, produce partial directional stop behavior, and give only moderate forward-transmission reductions in finite arrays. By contrast, connected low-thickness ligaments strengthen modal coupling and generate wider stop intervals along the sampled path. The crossed-strip and cross-connected circular networks reduce the full-aperture transmission at their selected stop-band frequencies. The cross-connected y-elongated elliptical network gives the strongest response. These results show that ABH topology, and especially the connectivity and anisotropy of the thinned regions, is a decisive design parameter for directional attenuation in lightweight periodic plates.