Spin relaxation of spin-1/2 quantum particles on linear chains with random Larmor frequencies is studied. Wave packets are prepared with initial spin polarization in the propagation direction and with either zero, or finite group velocity. The discrete Schrödinger-Pauli equation is solved numerically. Exponential spin relaxation is observed in a restricted time range. The decay constant depends on the width of the distribution of the Larmor frequencies and on the group velocity. No dependence on the initial halfwidth of the wave packet is found.