We report a direct determination of the dynamic behavior of confined acoustic phonons in nanocavities by picosecond acoustics. We provide the broadband, high resolution transmission amplitude curve in the subterahertz range, and we give evidence of resonant transmission peaks in three successive stop bands, in quantitative agreement with acoustic simulations. We furthermore demonstrate transit times in the nanosecond range at the cavity peaks reflecting the strong confinement of resonant phonons within the cavity layer. On the other hand, picosecond transit times are measured in the stop band, shorter than in any of the constituting materials, a tunneling effect well known both in photonic crystals and in macroscopic phononic systems.