We have used narrow emission-line data from the new 7C Redshift Survey to investigate correlations between the narrow-line luminosities and the radio properties of radio galaxies and steep-spectrum quasars. The 7C Redshift Survey is a low-frequency (151MHz) selected sample with a flux density limit about 25 times fainter than the 3CRR sample. By combining these samples, we can for the first time distinguish whether the correlations present are controlled by 151-MHz radio luminosity L151 or redshift z. We find unequivocal evidence that the dominant effect is a strong positive correlation between narrow-line luminosity LNLR and L151, of the form LNLR~L1510.79+/-0.04. Correlations of LNLR with redshift or radio properties, such as linear size or 151-MHz (rest frame) spectral index, are either much weaker or absent. We use simple assumptions to estimate the total bulk kinetic power Q of the jets in FRII radio sources, and confirm the underlying proportionality between jet power and narrow-line luminosity first discussed by Rawlings & Saunders. We make the assumption that the main energy input to the narrow-line region is photoionization by the quasar accretion disc, and relate Q to the disc luminosity, Qphot. We find that 0.05<~QQphot<~1, so that the jet power is within about an order of magnitude of the accretion disc luminosity. Values of QQphot~1 require the volume filling factor η of the synchrotron-emitting material to be of the order of unity, and in addition require one or more of the following: (i) an important contribution to the energy budget from protons; (ii) a large reservoir of mildly relativistic electrons; and (iii) a substantial departure from the minimum-energy condition in the lobe material. The most powerful radio sources are accreting at rates close to the Eddington limit of supermassive black holes (MBH>~109Msolar), whilst lower power sources are accreting at sub-Eddington rates.