The present paper reports on the modeling of linear elastic properties of acoustically insulating foams with unit cells containing solid films or membranes at the junction between interconnected pores from a numerical homogenization technique. It combines fluid-flow induced microstructure identification with simulations of the effective Young's modulus and Poisson ratio from a mixture of routinely available laboratory measurements (porosity, permeability, cell size) and finite element calculations when the boundary conditions of the periodic unit cell take particular symmetric forms. This combination results in microstructural determination of the macroscopic coefficients entering into the Biot-Allard theory of wave propagation and dissipation through porous media. Precise control over pore morphology and mechanical properties of the base material renders this multi-scale approach particularly suitable for various advanced applications.