Low-temperature vertical carrier transport in layered structures comprised of Si nanocrystals separated in the growth direction by angstrom-thick SiO2 layers exhibits entirely unexpected, well-defined resonances in conductivity. An unusual alternating current (ac) conductivity dependence on frequency and low magnetic field, negative differential conductivity, reproducible N-shaped switching and self-oscillations were observed consistently. The modeled conductivity mechanism is associated with resonant hole tunneling via quantized valence band states of Si nanocrystals. Tight-binding calculations of the quantum confinement effect for different Si nanocrystal sizes and shapes strongly support the tunneling model.