The common derivation of Rossby waves is based on the quasi-geostrophic approximation. A simple non-harmonic approximation for extratropical Rossby waves on the sphere is proposed, in which the meridional coordinate is a parameter instead of a continuous variable. It is shown that, in contrast to the quasi-geostrophic solution, to first order the meridional structure of these non-harmonic Rossby waves becomes irrelevant for determining the dispersion relation in this theory. The proposed approximation accurately reproduces numerical results obtained from runs of an ocean general circulation model initiated from several initial meridional structures and captures the latitudinal dependence of the phase speed of these waves. The proposed theory yields explicit expressions for the dispersion relation and for the meridional structure of the waves.