We search for anisotropies in the arrival directions of cosmic rays observed by the KASCADE-Grande air shower experiment. The analysis is based on public data of about 23.7 million events with reconstructed primary energies above 1 PeV. We apply a novel maximum-likelihood reconstruction method for the cosmic ray anisotropy, that compensates for spurious anisotropies induced by local detector effects. We find no evidence for a large-scale dipole anisotropy in the data, consistent with official results based on the conventional East-West derivative method. On the other hand, a subset of cosmic rays with median energy of 33 PeV shows strong evidence for a medium-scale feature with an angular diameter of 40 degrees. After accounting for the look-elsewhere effect, the post-trial significance of this medium-scale feature is at the level of 4sigma.