We study theoretically vibrational modes in a nanowire superlattice consisting of an alternate stacking of two cylindrical layers. We focus on azimuthally symmetric torsional modes and calculate the phonon dispersion relations analytically. We also derive simple expressions for the frequency gaps generated in the nanowire superlattice. Moreover, we calculate the transmittance of phonons propagating through a nanowire superlattice with the finite number of periods. Based on our calculated results, effects of the superlattice longitudinal confinement and the radial confinement are examined.