This paper aims to study the dispersion phenomena of acoustic waves propagating in a periodic poroelastic medium. At mesoscale, the poroelastic saturated media is modeled by using Biot theory. We will compare two computational procedures for estimating the effective phase velocities and attenuation of plane waves in the period poroelastic structure at the macroscopic scale. First, wave-based Bloch analysis was employed to derive a finite element formulation which yields a quadratic complex eigenvalue problem. The equivalent fast/slow compressional and shear wave modes may be recognized by analyzing the computed complex wave numbers. Second, we used the asymptotic homogenization method to derive the poroelastic and dynamic permeabilities properties of an effective poroelastic model which allows us to estimate the effective wave dispersion. The polarization of wave modes at the cell level may be reconstructed from macroscopic solution. Numerical results show that both methods could provide well-matched estimations of the effective phase velocities and attenuation within the first Brillouin zone associated with the periodic structure