Theoretical investigation of linear dichroism in the angular distribution (LDAD) of photoelectrons obtained in a molecular photoionization process is presented. The developed theory applies to the investigation of both molecules adsorbed on a surface and gas phase molecules when the molecular orientation is fixed by coincidence between a photoelectron and a residual ion resulting from fast molecular dissociation. Several experimental geometries are proposed which are used to unravel the properties of the adsorbed or oriented molecules. The first numerical calculations of LDAD in the case of photoionization of the 4 sigma-orbital of fixed-in-space CO molecules are presented. We calculate the differential cross section and the LDAD for two of the geometries of the experiment we have proposed : One allows a simple extraction of theoretical parameters from the experimental data in a kind of complete experiment with molecules; the other allows the determination of molecular orientation in space, for example, with respect to the surface if the nuclear motion of adsorbates is neglected.