In the present paper, the performance of Biot's theory is investigated for wave propagation in cellular and porous solids with entrained fluid for configurations with well-known drained (no fluid) mechanical properties. Cellular solids differ from porous solids based on their relative density $\rho *<0.3$. The distinction is phenomenological and is based on the applicability of beam (or plate) theories to describe microstructural deformations. The wave propagation in a periodic square lattice is analyzed with a finite-element model, which explicitly considers fluid-structure interactions, structural deformations, and fluid-pressure variations. Bloch theorem is employed to enforce symmetry conditions of a representative volume element and obtain a relation between frequency and wavevector. It is found that the entrained fluid does not affect shear waves, beyond added-mass effects, so long as the wave spectrum is below the pores' natural frequency. One finds strong dispersion in cellular solids as a result of resonant scattering, in contrast to Bragg scattering dominant in porous media. Configurations with $0.0001\u2264\rho *\u22641$ are investigated. One finds that Biot's theory, derived from averaged microstructural quantities, well estimates the phase velocity of pressure and shear waves for cellular porous solids, except for the limit $\rho *\u21921$. For frequencies below the first resonance of the lattice walls, only the fast-pressure mode of the two modes predicted by Biot's theory is found. It is also shown that homogenized models for shear waves based on microstructural deformations for drained conditions agree with Biot's theory.