The starting point of this paper is a model of the non-Fickian diffusion of a simple fluid into a polymeric medium that has been introduced in El Afif and Grmela (2002). The model is extended to a mixture consisting of two simple fluids and one polymeric medium. The effects of the polymeric medium on diffusion are included in the formulations through the use o relative momenta playing the role of internal variables. In ternary mixtures, new phenomena arise due to cross-coupling effects. As a consequence, the single diffusion Deborah number in binary mixtures becomes a 2 x 2 matrix in ternary mixtures. Two specific examples for which experimental data are available are investigated in detail.