Multiple-quantum (MQ) nuclear magnetic resonance (NMR) spin dynamics are investigated in rigid linear chains and rings with nearest neighbor dipole-dipole interactions with different coupling constants due to spatial disorder. It is shown that MQ NMR spectra, for such one-dimensional systems initially at thermal equilibrium followed by evolution under a 2-quantum/2-spin average dipolar Hamiltonian, only consist of 0- and 2-quantum coherences. A new constant of motion for the systems under consideration is found and used in the numerical analysis of MQ NMR spin dynamics to factorize the Hamiltonian into distinct blocks corresponding to different eigenvalues of the constant of motion. Only one of these blocks of dimension NxN (where N is the number of spins) completely determines the MQ NMR spin dynamics. Supercomputer calculations of MQ NMR spin dynamics in rigid linear chains containing up to 1000 spins are presented. The possibility to obtain structural information from the time evolution of MQ coherences is also discussed.