A transient three-dimensional Darcy model of natural convection in porous medium filled cavities is studied, using a vorticity-vector potential formulation and the generalized integral transform technique (GITT). A general formulation and solution methodology for vertical cavities (insulated vertical walls with differential horizontal wall temperatures) is developed. Results for cubic cavities are presented while evaluating the Rayleigh number effects for stable situations, observing the transient evolution of the heat transfer process. The convergence behavior of the proposed eigenfunction expansion solution is investigated and comparisons with previously reported steady-state solutions are critically performed.