The dynamical properties of uniform two-dimensional arrays of nickel nanowires have been investigated by inelastic light scattering. Multiple spin waves are observed that are in accordance with dipole-exchange theory predictions for the quantization of bulk spin waves. This first study of the spin-wave dynamics in ferromagnetic nanowire arrays reveals strong mode quantization effects and indications of a subtle magnetic interplay between nanowires. The results show that it is important to take proper account of these effects for the fundamental physics and future technological developments of magnetic nanowires.