Phononic crystals with triangular and honeycomb lattices are investigated experimentally and theoretically. They are composed of arrays of steel cylinders immersed in water. The measured transmission spectra reveal the existence of complete band gaps but also of deaf bands that cause strong attenuation in the transmission. Band gaps and deaf bands are identified by comparing band structure computations, obtained by a periodic- boundary finite element method, with transmission simulations, obtained using the finite difference time domain method. Using the same cylinders and the same distance between them, it is found that the center frequency of the first full band gap of the honeycomb lattice is lower than that of the triangular lattice, and that the width of this full band gap is also wider. Comparing our results with air/solid phononic crystals, we find that the fluid/solid system has larger band gaps than the air/solid system in the low frequency range.