Biological composites are known to exhibit extraordinary mechanical properties, due to their complex, hierarchical internal structures composed of fibrous polymers, and other supra-molecular arrangements. While much is known about their quasi-static mechanical properties, the study of elastic wave propagation in biological structural materials is an almost pristine field. In this work, we investigate the phononic behavior of the anisotropic fibrillar network in micron-thick onion cell walls using laser-generated, sub-GHz guided acoustic waves. The measured dispersion curves reveal the presence of Rayleigh and Sezawa modes. By tracking their dispersion as a function of cell orientation, we find angular-dependent phase velocities due to mechanical anisotropy of the wall. Moreover, we show that the structural arrangement of the wall generates an anomalous low-frequency acoustic branch characterized by an ultra-slow group velocity. We demonstrate that this mode arises from the hybridization of modes guided within the wall and the resonance of subcell structures. Our findings pave the way to the design of new biologically-derived acoustic metamaterials.