The theory of nuclear quadrupolar relaxation due to translational diffusion in crystalline solids is investigated. Unlike the case of dipolar relaxation, it is often necessary to take into account three-particle, as well as two-particle, probability functions in the expression for the spectral densities of the quadrupolar interaction. In particular, the nuclear relaxation of a particle diffusing on a cubic lattice, due to the relative translational motion of other identical particles on the same lattice, is evaluated according to the mean field theory which includes, in an average way, the effects of spatial correlations between the motions of particles. Formal expressions are obtained for the three-particle probability function for nearest-neighbour jumps on SC, FCC and BCC lattices. Analytical formulae are derived for the spectral density functions in the low-temperature limit and demonstrate the greater importance of three-particle effects as the concentration of particles increases. In a subsequent paper, the mean field theory is applied to the quadrupolar relaxation of deuterons in transition metal deuterides.