We report the detection of shock-excited far-infrared emission of H2O, OH, and CO from the supernova remnant 3C 391, using the Infrared Space Observatory Long-Wavelength Spectrometer. This is the first detection of thermal H2O and OH emission from a supernova remnant. For two other remnants, W28 and W44, CO emission was detected, but OH was detected only in absorption. The observed H2O and OH emission lines arise from levels within ~400 K of the ground state, consistent with collisional excitation in warm, dense gas created after the passage of the shock front through the dense clumps in the preshock cloud. The postshock gas we observe has a density ~2×105 cm-3 and temperature 100-1000 K, and the relative abundances of CO:OH:H2O in the emitting region are 100:1:15 for a temperature of 200 K. The presence of a significant column of warm H2O suggests that the chemistry has been changed significantly by the shock. The existence of significant column densities of both OH and H2O, which is at odds with models for nondissociative shocks into dense gas, could be due to photodissociation of H2O or a mix of fast and slow shocks through regions with different preshock density.