Motivated by the need to better understand the influence of the Maxwell-Cattaneo effect (or hyperbolic heat flow) on the dynamics of porous media in local thermal non-equilibrium, the stability of a porous Darcy-Brinkman layer is studied when the Maxwell-Cattaneo (MC) relation of temperature and heat flux is introduced to a fluid and solid. We first prove that, in the absence of the MC effect, the porous layer cannot support oscillatory motions. When the MC effect is present in the fluid only, propagation of oscillatory motions is possible, provided that the MC effect parameter exceeds a certain threshold. The oscillatory motions are then preferred only if the thermal interphase interaction parameter H is small. On the other hand, when the MC effect is present in the solid only, the oscillatory instability is enhanced when H is large. The contrasting influences of the MC effect on the fluid and solid lead to some novel features when the MC effect is present simultaneously in both fluid and solid. Here, oscillatory motions can be preferred for intermediate values of H, depending on the two MC parameters measuring the influences in the solid and fluid. Although the presence of the MC effect introduces new modes so that the frequency equation changes from linear in the frequency squared to cubic, the unstable mode is always provided by the mode which is stable in the absence of the MC effect made unstable by the presence of the MC effect. The new modes are never preferred, but they can possess Taken-Bogdanov's bifurcations in addition to the Hopf bifurcations present in all the cases. When the analysis is applied to crude oil in sandstone and water in sandstone, we find that they possess contrasting stability properties.