We present high-resolution optical echelle spectroscopy for a large fraction of the Li-rich late-type stars recently discovered in the vicinity of the Lupus dark clouds. Our results confirm the high Li i λ6708 equivalent widths previously estimated from medium-resolution spectra, thus adding strength to the conclusion that the large majority of these stars are still in the pre-main-sequence phase of their evolution, contrary to claims from other authors that many of them might be zero-age main-sequence stars. We present a statistical approach to derive a mean distance for the sample, and find that it is consistent with, or slightly lower than, the Hipparcos distance of the Lupus star-forming region. The radial velocities measured for part of these stars are consistent with those observed for the Lupus star-forming region, while stars outside the dark clouds show a mean difference of the order of 3 km s−1. The projected rotational velocities show a lack of slow rotators, which is interpreted as a consequence of the X-ray selection of the sample. The Li-rich stars in Lupus studied in this work yield a fairly ‘clean’ sample of very young stars, while in other star-forming regions a larger fraction of older zero-age main-sequence stars has been found among ROSAT-discovered Li-rich stars. We argue that this fact reflects the relation of these stars with the Gould Belt.