A microscopic theory of the Raman scattering based on the local bond-polarizability model is presented and applied to the analysis of phonon confinement in porous silicon and porous germanium, as well as nanowire structures. Within the linear response approximation, the Raman shift intensity is calculated by means of the displacement-displacement Green's function and the Born model, including central and non-central interatomic forces. For the porous case, the supercell method is used and ordered pores are produced by removing columns of Si or Ge atoms from their crystalline structures. This microscopic theory predicts a remarkable shift of the highest-frequency of first-order Raman peaks towards lower energies, in comparison with the crystalline case. This shift is discussed within the quantum confinement framework and quantitatively compared with the experimental results obtained from porous silicon samples, which were produced by anodizing p--type (001)-oriented crystalline Si wafers in a hydrofluoric acid bath.