A recent paper [A.A. Krokhin, J. Arriaga, L.N. Gumen, Speed of sound in periodic elastic composites, Phys. Rev. Lett. 91 (2004) 264302-1-4] addresses the speed of sound in periodic elastic composites (phononic crystals) with particular emphasis to the case where air bubbles are present in water and arranged periodically. In such periodically arranged mixtures, the well-known phenomena of the drop of the speed of sound may occur and applications related to, e.g., sound-beam focusing and acoustic surgery are possible [F. Cervera, L. Sanchez, J.V. Sanchez-Perez, R. Martinez-Sala, C. Rubio, F. Meseguer, C. Lopez, D. Caballero, J. Sanchez-Dehesa, Phys. Rev. Lett. 88 (2002) 023902]. In this paper, the analysis is extended theoretically to include cases where a background flow in a periodic structure is maintained. Calculations of dispersion relations and group velocities are presented in cases with one- and two-dimensional material periodicity for background flow values in the range: 0-1m/s. Materials considered in the calculations are periodic water-air mixtures. It is shown that acoustic waves couple to the group velocities only if the (acoustic) wave vector has a component along the background flow velocity direction.