<jats:p>The theory of dipolar nuclear magnetic relaxation in molecules with multiple internal motions is extended to the case where the motion of the molecule as a whole is anisotropic, with no restriction on the magnitudes of the correlation times involved. Numerical calculations of the 13C relaxation times T1 and T2 for 13C–1H relaxation in a hydrocarbon chain attached to an axially symmetric prolate ellipsoid are presented. Effects of the anisotropic motion of the molecule can be seen as far as five carbons along the chain, even when the motion about the C–C bonds in the chain is a factor of 10 faster than the fastest motion of the molecule as a whole. Two cases have been considered in which the motion about the long axis of the ellipsoid is fast [(ωC + ωH)2 / 6DZ2 ≪ 1] or slow [(ωC + ωH)2 / 6DZ2 &gt; 1] compared to the highest relevant Larmor frequency. The effects on T1 and T2 of changes in the axial ratio of the ellipsoid and in the angle between the first bond in the chain and the long axis of the ellipsoid, which are markedly different in these two cases, can be explained in terms of the effectiveness of the different motions in the system for relaxation. 13C spin-lattice relaxation times have been determined for all the resolved resonances in n -alkanes from C6 to C18 and in n -alkyl bromides from C4 to C15. Using the theoretical treatment developed in the first part of the paper, these relaxation times have been used to calculate the diffusion coefficients for the various motions of these molecules (considered as axially symmetric prolate ellipsoids with internal motion). In the shorter-chain n -alkanes (C6 to C10), where the resonances of all the carbons can be resolved, it is possible to calculate the two diffusion coefficients of the molecules as a whole and the diffusion coefficients for motion about each C–C bond in the chain. The diffusion coefficients about all the bonds in the chain (except that to the terminal methyl) are the same, being in the range 0.1−0.2 × 1011 sec−1 (about a factor of 10 slower than the motion of the whole molecule about its long axis). The motion about the terminal bond is a factor of 2–6 faster. In longer chain alkanes (C12–C18), where not all the resonances are resolved, only the diffusion coefficients about the four bonds nearest the terminal methyl can be obtained. Comparison of the present results with those obtained earlier with dipalmitoyl lecithin bilayers indicate that the region of the hydrocarbon chain near the terminal methyl group has essentially identical motional freedom in the bilayer and in the simple n -alkanes, while the region near the glycerol moiety in the bilayer is motionally much more restricted.</jats:p>