A method of the linear combinations of atomic orbitals for cubic crystals with an $s$ orbital on each atom is proposed, in which the Bloch wave functions (equivalent to the basis functions of the one-dimensional representations of the subgroup of lattice translations) are replaced by basis functions of the irreducible representation of the cubic point group. The coefficient functions of the atomic orbitals which enter into the new wave functions are solutions of the Wannier-Slater equation for a given type of lattice and for interactions between atomic neighbors. These functions meet the requirement that they vanish at the crystal boundary. For a pure crystal, the electron density need only be analyzed at one representative atomic site. Since a site like this can be put in the center of the system of coordinates, the needed coefficient functions can be the basis functions of only one irreducible representation, viz., that of the total symmetry of the cubic point group. They can be approximated in terms of a few cubic harmonics belonging to the irreducible representation mentioned and in terms of spherical Bessel functions equal in order to the cubic harmonics. Unlike the theory of Bloch, where the wave functions and energies depend on a three-component vector parameter, the present scheme introduces only one scalar parameter for the quantization of the electron states. This enables one to reduce the integration generating the Green's function to a one-dimensional one. The energies can be expressed as sums of powers of the quantum parameter and the band is obtained as a set of states which give the nonvanishing contribution to the electron density of the crystal. The band structure obtained from approximate solutions for the face-centered cubic lattice is compared with that obtained from Bloch's method. In the tight-binding approach these solutions and Bloch's method give the energy dependencies of the density of states which are close to each other over about two-thirds of the bandwidth. For almost-free electrons, these dependencies are nearly coincident within the interval of the energy between the band bottom and a certain level below the critical one in Bloch's band. With the same solutions we obtain bandwidths which are identical with Bloch's in the tight-binding case, but nearly double Bloch's in the case of almost-free electrons.