We present the results of a search for untriggered gamma-ray burst (GRB) afterglows with the Robotic Optical Transient Search Experiment-III (ROTSE- III) telescope array. This search covers observations from 2003 September to 2005 March. We have an effective coverage of 1.74 deg(2) yr for rapidly fading transients that remain brighter than similar to 17.5 mag for more than 30 minutes. This search is the first large-area survey to be able to detect typical untriggered GRB afterglows. Our background rate is very low and purely astrophysical. We have found four previously unknown cataclysmic variables (CVs) and one new flare star. We have not detected any candidate afterglow events or other unidentified transients. We can place an upper limit on the rate of fading optical transients with quiescent counterparts dimmer than similar to 20th magnitude at a rate of less than 1.9 deg(-2) yr(-1) with 95% confidence. This places limits on the optical characteristics of off-axis (orphan) GRB afterglows. As a by-product of this search, we have an effective similar to 52 deg(2) yr of coverage for very slowly decaying transients, such as CVs. This implies an overall rate of outbursts from high Galactic latitude CVs of 0.1 deg(-2) yr(-1).