Optical absorption of circularly polarized light is well known to yield an electron spin polarization in direct band gap semiconductors. We demonstrate that electron spins can even be generated with high efficiency by absorption of linearly polarized light in InxGa(1-x)As. By changing the incident linear polarization direction we can selectively excite spins in both polar and transverse directions. These directions can be identified by the phase during spin precession using time-resolved Faraday rotation. We show that the spin orientations do not depend on the crystal axes suggesting an extrinsic excitation mechanism.