The spectrum and damping of waves in partially randomized multilayer structures are calculated. A method of calculation that was proposed and demonstrated earlier, for the model of a superlattice with a harmonic dependence of its material parameters along its axis in the initial state, is extended to the case of a multilayer structure (i.e., a superlattice with sharp interfaces). One- and three-dimensional random modulations of the period are considered, and the correlation function of the superlattice is derived as a series in which each term is a product of a harmonic and a monotonically decaying function. The law of decay of the correlation function is Gaussian for smooth inhomogeneities, and has different forms for one- and three-dimensional short-wavelength inhomogeneities. The spectrum and damping of waves in the superlattice described by this correlation function are found in the weak-coupling approximation in the vicinities of all of the odd Brillouin zone boundaries. Analytical dependences of the main characteristics of the spectrum and damping on the zone number n are obtained. The conditions for the closing of the gaps at the Brillouin zone boundaries are derived, and depend on the dimensionality of the inhomogeneities and the degree of their smoothness.