Shift and injection currents are generated in the wurtzite semiconductors CdSe and CdS at 295 K using above-band-gap (ℏω≫Eg) femtosecond pulses and detected via the emitted terahertz radiation; the optical beams are normally incident on samples with the optic axis in the plane of the surface. For optical intensities up to 75 MW cm-2 (or carrier density ≪1018 cm-3) the terahertz radiation amplitude shows the expected linear dependence and also varies with optical polarization and sample orientation consistent with the third-rank tensors that govern the current generation processes in the wurtzite structure. The largest shift currents are generated along the optical axis for light polarized along that axis. In CdSe with ℏω=1.80 eV (690 nm), the electron shift distance is ∼40% of the 0.25 nm bond length and the peak current density is 5 kA cm-2 for an optical intensity of 10 MW cm-2; for CdS the corresponding experiment at ℏω=3.0 eV (410 nm) gives a shift distance ∼80% of the 0.26 nm bond length with a peak current density of 50 kA cm-2 for an - incident intensity of 75 MW cm-2. For injection current produced in CdSe with circularly polarized 690 nm excitation, electrons are injected with an average speed of 9 km s-1; this is ∼3% of the group velocity for electrons excited with the same energy. The corresponding values for CdS excited at 410 nm are 20 km s-1 and 2%. From the temporal characteristics of the terahertz emission for injection currents in CdS we deduce that the electron momentum scattering time is ≪100 fs, consistent with mobility studies.