The South Pole Telescope (SPT) is conducting a Sunyaev-Zeldovich (SZ) effect survey over large areas of the southern sky, searching for massive galaxy clusters to high redshift. In this preliminary study, we focus on a 40 square-degree area targeted by the Blanco Cosmology Survey (BCS), which is centered roughly at right ascension 5h30m, declination -53 degrees. Over two seasons of observations, this entire region has been mapped by the SPT at 95 GHz, 150 GHz, and 225 GHz. We report the four most significant SPT detections of SZ clusters in this field, three of which were previously unknown and, therefore, represent the first galaxy clusters discovered with an SZ survey. The SZ clusters are detected as decrements with greater than 5-sigma significance in the high-sensitivity 150 GHz SPT map. The SZ spectrum of these sources is confirmed by detections of decrements at the corresponding locations in the 95 GHz SPT map and non-detections at those locations in the 225 GHz SPT map. Multiband optical images from the BCS survey demonstrate significant concentrations of similarly colored galaxies at the positions of the SZ detections. Photometric redshift estimates from the BCS data indicate that two of the clusters lie at moderate redshift (z ~ 0.4) and two at high redshift (z >~ 0.8). One of the SZ detections was previously identified as a galaxy cluster using X-ray data from the ROSAT All-Sky Survey (RASS). Potential RASS counterparts (not previously identified as clusters) are also found for two of the new discoveries. These first four galaxy clusters are the most significant SZ detections from a subset of the ongoing SPT survey. As such, they serve as a demonstration that SZ surveys, and the SPT in particular, can be an effective means for finding galaxy clusters.