Using two identical telescopes at widely separated longitudes, the ROTSE-III network observed decaying emission from the remarkably bright afterglow of GRB 030329. In this report we present observations covering 56% of the period from 1.5-47 hours after the burst. We find that the light curve is piecewise consistent with a powerlaw decay. When the ROTSE-III data are combined with data reported by other groups, there is evidence for five breaks within the first 20 hours after the burst. Between two of those breaks, observations from 15.9-17.1 h after the burst at 1-s time resolution with McDonald Observatorys 2.1-m telescope reveal no evidence for fluctuations or deviations from a simple power law. Multiple breaks may indicate complex structure in the jet. There are also two unambiguous episodes at 23 and 45 hours after the burst where the intensity becomes consistent with a constant for several hours, perhaps indicating multiple injections of energy into the GRB/afterglow system.