A two-dimensional porous coordination polymer (NH4)2(HOOC(CH2)4COOH)[Zn2(C2O4)3] (abbreviated as (NH4)2(adp)[Zn2(ox)3] (adp = adipic acid, ox = oxalate)), which accommodates water molecules between the [Zn2(ox)3] layers, is highly remarked as a new type of crystalline proton conductor. In order to investigate its phase behavior and the proton conducting mechanism, we have performed adiabatic calorimetry, neutron diffraction, and quasi-elastic neutron scattering experiments on a fully hydrated sample (NH4)2(adp)[Zn2(ox)3]·3H2O with the highest proton conductivity (8 × 10-3 S cm-1, 25 °C, 98% RH). Its isostructural derivative K2(adp)[Zn2(ox)3]·3H2O was also measured to investigate the role of ammonium ions. (NH4)2(adp)[Zn2(ox)3]·3H2O and K2(adp)[Zn2(ox)3]·3H2O exhibit higher order transitions at 86 K and 138 K, respectively. From the magnitude of the transition entropy, the former is of an order-disorder type while the latter is of a displacive type. (NH4)2(adp)[Zn2(ox)3]·3H2O has four Debye-type relaxations and K2(adp)[Zn2(ox)3]·3H2O has two similar relaxations above each transition temperature. The two relaxations of (NH4)2(adp)[Zn2(ox)3]·3H2O with very small activation energies (ΔEa < 5 kJ mol-1) are due to the rotational motions of ammonium ions and play important roles in the proton conduction mechanism. It was also found that the protons in (NH4)2(adp)[Zn2(ox)3]·3H2O are carried through a Grotthuss mechanism. We present a discussion on the proton conducting mechanism based on the present structural and dynamical information.