The Australia Telescope Compact Array (ATCA) has been used at 1.38 and 2.38GHz to survey seven southern Abell clusters of galaxies with high X-ray luminosities: A2746, A2837, A3126, A3216, A3230, A3827 and A3836. The clusters have also been surveyed at 0.843GHz with the Molonglo Observatory Synthesis Telescope (MOST). We have listed a complete 1.38-GHz sample of 149 radio sources within the Abell circles centred on their X-ray centroids. We compare their identification fractions, emitted 1.38-GHz and optical powers, radio spectral indices and radial variation in projected source density with those of the radio-selected samples of Slee et al. (1998AuJPh..51..971S). We compare our fractional radio luminosity function with that of the radio-selected samples of Ledlow and Owen (1996AJ....112....9L) and Slee et al. (1998AuJPh..51..971S). Three significant differences are noted between X-ray and radio-selected samples of clusters; (1) the X-ray sample has an excess of flat-spectrum radio sources; (2) the fractional radio luminosity function for the FR I sources in the X-ray selected sample is much steeper, implying that fewer of their cluster galaxies become hosts for the stronger FR I radio galaxies; (3) a complete absence of FR II radio galaxies in the X-ray selected sample. The average excess projected density of radio sources near our cluster centres is approx. 5 times the background source density.