The spin-lattice relaxation rates of H-1 and K-39 nuclei in KHSeO4 crystals were studied in the temperature range 160-400 K. The spin-lattice relaxation recovery of H-1 nucleus in this crystal can be represented with a single exponential function, and the relaxation T-1(-1) curve of H-1 can be represented with the Bloembergen-Purcell-Pound (BPP) function. The relaxation process of K-39 with dominant quadrupole relaxation can be described by a linear combination of two exponential functions. T-1(-1) for the K-39 nucleus was found to have a very strong temperature dependence, T-1(-1) = beta T-7. Rapid variations in relaxation rates are associated with critical fluctuations in the electronic spin system. The T-7 temperature dependence of the Raman relaxation rate is shown here to be due to phonon-magnon coupling. (c) 2005 Elsevier Ltd. All rights reserved.