This study focuses on how Lagrangian coherent structures (LCSs) control solute dispersion in heterogeneous poroelastic media (HPM). We show how interactions between medium compressibility, conductivity heterogeneity, and periodic forcing give rise to complex flows and diverse LCS types (KAM islands, chaotic saddles, etc) that have profound impacts on diffusive solute transport (main image) that do not arise in the steady counterpart (inset). Strongly anomalous transport impacts both spatial moments and residence time distributions and persists at low P\'eclet numbers. This study reveals the complex transport phenomena that can arise in HPM and shows how LCSs govern solute dispersion.