Spin-lattice relaxation times of P1 centers in a suite of two natural type Ib, two synthetic type Ib, and one natural type Ia diamonds were measured at 9.6 GHz as a function of temperature in the range 300 K> T> 4.2 K. An analysis of the results revealed that for three of the diamonds (two synthetic type Ib and the natural type Ia) spin-orbit phonon-induced tunneling is the main relaxation mechanism. In the case of the Ia diamond cross-relaxation takes place between P1 and P2 centers. In the natural type Ib samples a much more effective relaxation mechanism dominates at lower temperatures. Electron spin resonance spectra of the latter samples revealed the presence of N3 centers. It seems that the more effective relaxation mechanism is associated with the N3 centers and that the P1 centers relax via the N3 centers to the lattice at these temperatures.