Non-Fickian behaviors in mass transport are very common in polymeric materials. The classical Fickian law, dictating the instantaneous response of the mass flux to changes in the driving force (namely the concentration gradient), seems to be too simplistic for mass transport in viscoelastic materials, generally linked through characteristic times to stress relaxation or structural changes. A constitutive equation in which the evolution of the mass flux is governed by a characteristic relaxation time thus appeared to be the most natural choice in order to catch anomalous behaviors. This has led since the '80s to the hyperbolic formulation of the transport equations, known as Cattaneo-Maxwell equations in the one-dimensional case. However, hyperbolic equations use for mass transport applications were limited by the well known but never understood the unpredictable appearance of inconsistencies (read unphysical concentration overshoots and negative concentration values). Recently, inconsistencies are explained by letting the hyperbolic formulation of transport equations in polymeric systems be derived from the theory of stochastic processes possessing finite propagation velocity, namely the Poisson-Kac processes. In this paper, the hyperbolic transport model based on partial probability waves is introduced and discussed. The model was applied to two relevant literature cases, showing to be suitable for the description of anomalous diffusional phenomena.