A theory of magnetic nuclear relaxation, providing a calculation of the correlation functions of complex motion of methyl groups is presented. The complex motion consists of jumps over the barrier (classical motion) and jumps through the barrier (tunneling). The Schrodinger equation has been applied in the calculation of the rate constant of tunneling jumps through the barrier. The equations for the spectral densities J(is)(m)(omega)), where omega = (omega +/- omega(T)), and omega = (momega(1)), where m = 1 or 2, omega(1) is the resonance angular frequency, and omega(T) is the angular frequency of the tunnel splitting, are derived. These spectral densities concern the motion of spin pairs inside methyl groups ("intra") and outside methyl groups ("inter"). The calculated spectral densities are applied to analyze the temperature dependencies of the spin-lattice relaxation rate in solids containing methyl groups. A wide regime of temperatures from 0 K up to the melting point is considered. The temperature at which the tunneling process ceases is discussed. The theory proposed explains the different temperature dependencies of T-1 for CH3COOK obtained in the experiments caused by small amounts of CH3COOH or water impurities. The theoretical equations derived in this paper are compared to those known in the literature.