The orientation and frequency dependence of nuclear spin relaxation rates can provide detailed information about the amplitudes and rates of collective orientational fluctuations (director fluctuations) in liquid crystals. In particular, the low-frequency spin relaxation rates from a lamellar phase reflect the membrane bending rigidity and the intermembrane forces. This information is contained in three spectral density functions J(n)(omega), n=0,1,2. We have recently presented a continuum-mechanical theory for the second-order spectral density J(1)(omega). Here we extend the theory to the fourth-order spectral densities J(0)(omega) and J(2)(omega), which dominate the transverse relaxation rate in the parallel and perpendicular configurations. These spectral densities have previously been considered in connection with director fluctuations in nematic phases, neglecting the elastic and hydrodynamic anisotropy of the phase. In lamellar phases, this anisotropy plays a crucial role and must be retained in the relaxation theory. Director fluctuations can be induced by elastic distortion modes as well as by molecular translational diffusion. In a lamellar phase, these independent processes give rise to qualitatively different spin relaxation behavior. In particular, J(0)(0) and J(2)(0) diverge in the limit of quenched disorder. The theoretical results presented here are directly applicable to spin relaxation data from a variety of lamellar systems, including phospholipid bilayers and sterically stabilized dilute lamellar phases. An analysis of published H-2 and P-31 relaxation data from phospholipid bilayer phases is presented, leading to a qualitatively different picture, from what has previously been deduced in terms of a free membrane theory.