The dynamical current response of a multiple-quantum-wire superlattice to a time-oscillatory external electric field is examined using Kubo's linear-response theory. Couplings of the quasi-one-dimensional electron gases with phonons (polar optic, and acoustic deformation potential) are discussed to lowest order in terms of a current-current correlation function which embodies both intrawire and interwire electron-electron Coulomb interactions. The associated dynamic screening of the electron-phonon interaction is approximated by employing the random-phase-approximation density-density correlation function for the interacting electrons of the quantum-wire superlattice. The frequency-dependent conductivity of this system is evaluated numerically, exhibiting its variations with temperature and geometrical superlattice parameters.