We investigated theoretically and experimentally the acoustic waves tunneling and resonant-tunneling through one-dimensional phononic crystals within the frequency gap. The phononic crystal consists of parallel and periodically arrayed plastic plates immersed in water. Acoustic tunneling was observed for this single crystal structure at frequencies inside the band gaps. By slightly increasing the separation of the two plates in the middle, a double-crystal structure forms, which allows acoustic waves to resonantly tunnel through it, with transmission peaks observed inside the band gaps. At resonance, the measured group time is large, while off resonance very fast group velocity is found.