The inverse relaxation time tensor in quasi-two-dimensional systems is calculated for elastic electron scattering by both acoustic and optical phonons. It is shown that in the general case the relaxation time is anisotropic. Also, on the basis of the Boltzmann equation in the anisotropic relaxation time approximation the electrical conductivity in these systems is calculated. It is found that anisotropy of the conductivity is defined not only by the anisotropy of effective mass and the relaxation time but also by the ratio of the Fermi level to the one-dimensional conduction band half-width. A comparison with available experimental data has been carried out. Qualitative agreement of theory with experiment was obtained.