Because of their tiny band gaps Dirac materials promise to improve the sensitivity for dark matter particles in the sub-MeV mass range by many orders of magnitude. We study several candidate materials and calculate the expected rates for dark matter scattering via light and heavy dark photons as well as for dark photon absorption. A particular emphasis is placed on how to distinguish a dark matter signal from background by searching for the characteristic daily modulation of the signal, which arises from the directional sensitivity of anisotropic materials in combination with the rotation of Earth. We revisit and improve previous calculations and propose two new candidate Dirac materials: bis(naphthoquinone)tetrathiafulvalene (BNQ-TTF) and Yb3PbO. We perform detailed calculations of the band structures of these materials and of ZrTe5 based on density functional theory and determine the band gap, the Fermi velocities, and the dielectric tensor. We show that in both ZrTe5 and BNQ-TTF the amplitude of the daily modulation can be larger than 10% of the total rate, allowing us to probe the preferred regions of parameter space even in the presence of sizable backgrounds. BNQ-TTF is found to be particularly sensitive to small dark matter masses (below 100 keV for scattering and below 50 meV for absorption), while Yb3PbO performs best for heavier particles.