We have fabricated ZnSeTe sinusoidal superlattices, i.e., periodic structures in which the concentration of Se and Te varies sinusoidally along the direction of growth. This was accomplished by means of a novel molecular-beam epitaxy growth technique in which the periodic modulation is achieved by substrate rotation in the presence of nonuniform flux distributions of Se and Te, rather than by shutter openings and closings. Superlattices fabricated by this technique indeed result in remarkably regular sinusoidal (or very nearly sinusoidal) compositional profiles, as inferred from the presence of only the first superlattice harmonics in the x-ray spectra observed on these novel structures. In addition to the structural analysis, we present extensive photoluminescence data which also provide clear evidence for the formation of a superlattice by this method of growth. Finally, we discuss the special advantages of such a shutterless method of superlattice fabrication, and the consequences of commensurability or noncommensurability of the superlattice period with the underlying crystal structure.