We present Infrared Space Observatory spectroscopy of sites in the supernova remnants W28, W44, and 3C391, where blast waves are impacting molecular clouds. Atomic fine-structure lines were detected from C, N, O, Si, P, and Fe. The S(3) and S(9) lines of H2 were detected for all three remnants. The observations require both shocks into gas with moderate (~ 100 /cm3) and high (~10,000 /cm3) pre-shock densities, with the moderate density shocks producing the ionic lines and the high density shock producing the molecular lines. No single shock model can account for all of the observed lines, even at the order of magnitude level. We find that the principal coolants of radiative supernova shocks in moderate-density gas are the far-infrared continuum from dust grains surviving the shock, followed by collisionally-excited [O I] 63.2 and [Si II] 34.8 micron lines. The principal coolant of the high-density shocks is collisionally-excited H2 rotational and ro-vibrational line emission. We systematically examine the ground-state fine structure of all cosmically abundant elements, to explain the presence or lack of all atomic fine lines in our spectra in terms of the atomic structure, interstellar abundances, and a moderate-density, partially-ionized plasma. The [P II] line at 60.6 microns is the first known astronomical detection. There is one bright unidentified line in our spectra, at 74.26 microns. The presence of bright [Si II] and [Fe II] lines requires partial destruction of the dust. The required gas-phase abundance of Fe suggests 15-30% of the Fe-bearing grains were destroyed. The infrared continuum brightness requires ~1 Msun of dust survives the shock, suggesting about 1/3 of the dust mass was destroyed, in agreement with the depletion estimate and with theoretical models for dust destruction.