The highly anisotropic nature of the band structure of the known organic metals determines whether the dominant spin-relaxation process in these compounds is of spin-spin or spin-phonon type. Tetrathiafulvalene (TTF) halides and pseudohalides are shown to be a family of compounds in which the spin-relaxation mechanism has a dipolar origin in contrast to the spin-phonon origin observed in the TTF-tetracyanoquinodimethane (TTF-TCNQ) family. This difference between the mechanisms of the relaxation is shown to be related to different anisotropies in the band structure, that of the halides and pseudohalides being more one-dimensional. Evaluation of the upper limit of the interstack hopping rate for TTF-Br0.76 from the multiple absorption lines observed in EPR at low temperatures and comparing it with the interstack hopping rate in TTF-TCNQ confirms this picture. The temperature dependence of the EPR linewidth measured for TTF-Br0.76 is understood in terms of the temperature dependence of the density of states at the Fermi level. The variation of the linewidth within the family of TTF halides from 9 to 200 G is explained in terms of variation of the interstack coupling-the larger the linewidth the smaller is the coupling. The metal-insulator transition temperatures are shown to be higher for compounds with the smaller interstack coupling, namely, those having a more one-dimensional band structure. © 1978 The American Physical Society.