Indirect detections of massive - presumably Jupiter-like - planets orbiting nearby Sun-like stars have recently been reported,. Rocky, Earth-like planets are much more difficult to detect, but clues to their possible existence can nevertheless be obtained from observations of the circumstellar debris disks of dust from which they form. The presence of such disks has been inferred from excess far-infrared emission but, with the exception of β Pictoris, it has proved difficult to image these structures directly as starlight dominates the faint light scattered by the dust. A more promising approach is to attempt to image the thermal emission from the dust grains at submillimetre wavelengths,. Here we present images of such emission around Fomalhaut, β Pictoris and Vega. For each star, dust emission is detected from regions comparable in size to the Sun's Kuiper belt of comets. The total dust mass surrounding each star is only a few lunar masses, so any Earth-like planets present must already have formed. The presence of the central cavity, approximately the size of Neptune's orbit, that we detect in the emission from Fomalhaut may indeed be the signature of such planets.