Photosynthetic organisms use networks of chromophores to absorb and deliver solar energy to reaction centers. We present a detailed model of the light-harvesting complexes in purple bacteria, including explicit interaction with sunlight, radiative and nonradiative energy loss, and dephasing and thermalizing effects of coupling to a vibrational bath. We capture the effect of slow vibrations by introducing time-dependent disorder. Our model describes the experimentally observed high efficiency of light harvesting, despite the absence of long-range quantum coherence. The one-exciton part of the quantum state fluctuates continuously but remains highly mixed at all times. These results suggest a relatively minor role for structure in determining efficiency. We build hypothetical models with randomly arranged chromophores but still observe high efficiency when nearest-neighbor distances are comparable to those in nature. This helps explain the high transport efficiency in organisms with widely differing antenna structures and suggests new design criteria for artificial light-harvesting devices.