Measurements of linear polarization are reported for thermally excited CO, /sup 13/CO, HCN, and CS in 14 dense clouds which encompass a broad range of physical conditions, gas dynamics, and possible magnetic field structure. Observing techniques are described which are sensitive to polarizations as small as 0.5%, and upper limits this small were derived in some cases. In clouds with magnetic fields, significant linear polarization has been predicted due to non-LTE population of the magnetic sublevels in each rotational state. No definite linear polarization was detected in any source. Assuming that the photon escape probability is generally anisotropic in these clouds, the lack of polarization could be the effect of unresolved kinematic or magnetic field structure.