High-resolution spectral hole-burning studies of CdSe/ZnS core/shell nanorods reveal a sharp zero-phonon line, with a line width dependent on the measurement time scale. The zero-phonon line width is attributed to contributions from radiative decay, spectral diffusion induced by surface electric field fluctuations, and phonon-assisted migration of excitons localized in the nanorods. A decoherence rate as small as 4.5 GHz has been observed, when the effects of spectral diffusion are suppressed in the spectral hole-burning measurement. Comparison between zero-phonon line widths in nanorods and spherical nanocrystals also elucidates important differences in the decoherence process between the one- and zero-dimensional nanostructures.