We calculate the dispersion of exciton-polariton modes in one-dimensional arrays of microcavities. We consider a two-sublattice array and a one-sublattice array of unevenly spaced spherical microcavities some of which contain embedded quantum dots. In both cases the dispersion of polariton eigen modes is shown to be efficiently controlled by a weak applied uniform strain. The structures we consider sustain slow-light modes that are of a specific importance for quantum optics applications. The high sensitivity of the optical response of microcavity arrays to the applied strain may be used in classical and quantum optical switches and the polariton based integrated photonics.