A theoretical framework is presented for the analysis of orientation and frequency dependent spin-relaxation rates from nuclei-diffusing on biaxial rodlike aggregates in rectangular and other amphiphilic liquid crystals. With the aid of eigenfunction expansion and integral transform techniques, the problem of calculating the spectral density functions for surface diffusion on a biaxial rod can be reduced to quadrature, without the need to specify the rod geometry. The general results are applied to ribbonlike and elliptic rods, yielding analytical results. The calculations demonstrate that spin relaxation is a considerably more sensitive probe of interface curvature than the static line shape. The relaxation anisotropy, in particular, can discriminate between closely similar rod geometries with different curvature distributions.