Photosynthetic light-harvesting proceeds by the collection and highly efficient transfer of energy through a network of pigment-protein complexes. Interchromophore electronic couplings and interactions between pigments and the surrounding protein determine energy levels of excitonic states, and dictate the mechanism of energy flow. The excitonic structure (orientation of excitonic transition dipoles) of pigment-protein complexes is generally deduced indirectly from x-ray crystallography, in combination with predictions of transition energies and couplings in the chromophore site basis. We demonstrate that coarse-grained, excitonic, structural information in the form of projection angles between transition dipole moments can be obtained from the polarization-dependent, two-dimensional electronic spectroscopy of an isotropic sample, particularly when the nonrephasing or free polarization decay signal, rather than the photon echo signal, is considered. This method provides an experimental link between atomic and electronic structure, and accesses dynamical information with femtosecond time resolution. In an investigation of the Fenna-Matthews-Olson complex from green sulfur bacteria, the energy transfer connecting two particular exciton states in the protein was isolated as the primary contributor to a crosspeak in the nonrephasing two-dimensional spectrum at 400 femtoseconds under a specific sequence of polarized excitation pulses. The results suggest the possibility of designing experiments using combinations of tailored polarization sequences to separate and monitor individual relaxation pathways.