The temperature dependencies of the nanosecond multiphonon relaxation (MR) rates of the 3F3 state of Tm3+ in the YLF crystal and of the 5F5 state of Ho3+ ion in the YAG and LuAG crystals and of the microsecond MR rates of the 4F9/2(2H9/2) state of Er3+ ions in YLF were measured in the wide temperature range using direct laser excitation and selective fluorescence kinetics decay registration. For YLF the observed relations are explained by 4-phonon process in the frame of a single-frequency model with hωeff=450 ± 30cm-1 for the 3F3 state of Tm3+ and by 5-phonon process with hωeff=445cm-1 for the 4F9/2(2H9/2) state of Er3+. For YAG and LuAG crystals these dependencies are explained by the 3-phonon process with hωeff=630cm-1. The decrease of the relaxation rate with the temperature in the range from 13 to 80 K was observed for the 4F9/2(2H9/2) state of Er3+ in the YLF crystal. It is explained by the redistribution of excited electronic states population of erbium ions over the higher lying Stark levels with different MR probabilities. A good fit of experimental temperature dependence (including the dropping part of the experimental curve) was obtained for single-frequency model (hωeff=450cm-1) with W01=8.0 × 104s-1 and W02=4.7 × 104s-1 accounting Boltzmann distribution of population over two excited Stark levels of the excited state of erbium ions. Employment of this model improves the fit between the experiment and the theory for the 5F5 state of Ho3+ ion in YAG as well. Strong influence of the parameters of the non-linear theory of MR, i.e. the reduced matrix elements U(k) of electronic transitions and the phonon factor of crystal matrix η on the spontaneous MR rates was observed experimentally. The smaller these parameters the slower the spontaneous MR W0. This fact can be used for searching new active crystal laser media for the mid-IR generation. © 2002 Elsevier Science B.V. All rights reserved.