We report the detection of hard X-ray emission components in the spectra of six nearby, giant elliptical galaxies observed with the ASCA satellite. The systems studied, which exhibit strong dynamical evidence for supermassive black holes in their nuclei, are M87, NGC 1399 and NGC 4696 (the dominant galaxies of the Virgo, Fornax and Centaurus clusters, respectively) and NGC 4472, 4636 and 4649 (three further giant ellipticals in the Virgo cluster). The ASCA data for all six sources provide clear evidence for hard emission components, which can be parametrized by power-law models with photon indices in the range Γ=0.6–1.5 (mean value 1.2) and intrinsic 1–10 keV luminosities of 2×1040–2×1042 erg s−1. Our results imply the identification of a new class of accreting X-ray source, with X-ray spectra significantly harder than those of binary X-ray sources, Seyfert nuclei or low-luminosity active galactic nuclei, and bolometric luminosities relatively dominated by their X-ray emission. We discuss various possible origins for the hard X-ray emission and argue that it is most likely to be due to accretion on to the central supermassive black holes, via low radiative efficiency accretion flows coupled with strong outflows. In the case of M87, our detected power-law flux is in good agreement with a previously reported measurement from ROSAT High Resolution Imager observations, which were able to resolve the jet from the nuclear X-ray emission components. We confirm previous results showing that the use of multiphase models in the analysis of the ASCA data leads to determinations of approximately solar emission-weighted metallicities for the X-ray gas in the galaxies. We also present results on the individual element abundances in NGC 4636.