We report the results of a VLA survey at 20cm of a complete sample of nearby Abell clusters (D<=3). We have compiled an extensive catalog of cluster radio sources with S_20_>10mJy from a combination of our VLA observations and observations of similar resolution with the Westerbork Synthesis Radio Telescope (WSRT). Using these radio data and X-ray data from the Einstein observatory, we have analyzed the properties of these clusters. Unlike that previously suggested, no significant correlation is found between 20cm radio power and X-ray luminosity. There is weak evidence that richer clusters have a higher probability of radio emission. Based on Rood-Sastry morphological types of clusters, we do not find any strong evidence that regular-type clusters are statistically more likely to be radio loud than irregular-type clusters. However, Bautz-Morgan types I and II-III are found to have comparable radio-detection rates; but Bautz-Morgan Type III clusters are likely to have a lower radio-detection rate. A weak correlation between X-ray-cooling mass-accretion rates and radio powers was found for central, dominant galaxies in cooling-flow clusters. This might suggest that either cooling accretion directly fuels the central engine and/or cooling flows strongly interact with (e.g., confine) the radio plasma. In addition, radio sources associated with central dominant galaxies in cooling-flow clusters tend to be small in comparison with those in non-cooling-flow clusters. This might suggest that cooling flows tend to obstruct the propagation of radio jets from central, dominant galaxies.