Directional detection of Dark Matter particles (DM) in the MeV mass range could be accomplished by studying electron recoils in large arrays of parallel carbon nanotubes. In a scattering process with a lattice electron, a DM particle might transfer sufficient energy to eject it from the nanotube surface. An external electric field is added to drive the electron from the open ends of the array to the detection region. The anisotropic response of this detection scheme, as a function of the orientation of the target with respect to the DM wind, is calculated, and it is concluded that no direct measurement of the electron ejection angle is needed to explore significant regions of the light DM exclusion plot. A compact sensor, in which the cathode element is substituted with a dense array of parallel carbon nanotubes, could serve as the basic detection unit.