We present Submillimetre Common-User Bolometer Array (SCUBA) photometry of a sample of eight high-redshift (2.5 less than or equal toz < 3.5) radio quasars from two redshift surveys: the TexOx-1000 (or TOOT) Survey and the 7C Redshift Survey (7CRS). Unlike the powerful high-redshift radio sources observed previously in the submillimetre, these radio sources are typical of those dominating the radio luminosity density of the population. We detect just two of the TOOT/7CRS targets at 850 mum, and one of these detections is probably due to synchrotron emission rather than dust. The population represented by the other six objects is detected in a statistical sense with their average 850-mum flux density implying that they are similar to low-redshift, far-infrared luminous quasars undergoing at most moderate (less than or similar to200 M(.)yr(-1)) starbursts. By considering all the SCUBA data available for radio sources, we conclude that positive correlations between rest-frame far-infrared luminosity L-FIR, 151-MHz luminosity L-151 and redshift z, although likely to be present, are hard to interpret because of subtle selection and classification biases, small number statistics and uncertainties concerning synchrotron contamination and k-correction. We argue that there is not yet any compelling evidence for significant differences in the submillimetre properties of radio-loud and radio-quiet quasars at high redshift.