The lattice dynamics of solid 4 He has been explored using pulsed NMR methods to study the motion of 3 He impurities in the temperature range where experiments have revealed anomalies attributed to superflow or unexpected viscoelastic properties of the solid 4 He lattice. We report the results of measurements of the nuclear spin-lattice and spin-spin relaxation times that measure the fluctuation spectrum at high and low frequencies, respectively, of the 3 He motion that results from quantum tunneling in the 4 He matrix. The measurements were made for 3 He concentrations 16 < x 3 < 2000 ppm. For 3 He concentrations x 3 = 16 ppm and 24 ppm, large changes are observed for both the spin-lattice relaxation time T 1 and the spin-spin relaxation time T 2 at temperatures close to those for which the anomalies are observed in measurements of torsional oscillator responses and the shear modulus. These changes in the NMR relaxation rates were not observed for higher 3 He concentrations.