The migration of organic compounds in stratified media is of fundamental concern in environmental and chemical engineering research. The diffusive transport of volatile organic compounds through laminate systems is characterized by partitioning, i.e., the development of concentration discontinuities at the interfaces between the individual laminae. If the transport is governed by cyclic transients, the relevant equations can be written in terms of coupled systems of diffusion equations subject to sinusoidal boundary conditions. This paper solves these systems of equations to present new algebraic solutions for propagation of the sinusoidal modes through arbitrary (finite) numbers of contiguous one-dimensional laminae. Both Cartesian and radial coordinate systems are considered. Two independent formulations of the lamina interface matching conditions are considered, corresponding to (1) an instantaneous partitioning model and to (2) a mass-limited partitioning model. It is shown that sinusoidal components of the concentration solutions propagate dispersively throughout the laminates. This is manifested, for example, in changes in shape of concentration pulses as measured at different points in the laminate system. Algorithms for generating exact dispersion relations for partitioning laminates are given, and an experimental technique for studying interfacial dynamics via frequency domain measurements is proposed.