We establish a Weitzenböck formula for harmonic morphisms between Riemannian manifolds and show that under suitable curvature conditions, such a map is totally geodesic. As an application of the Weitzenböck formula we obtain some non-existence results of a global nature for harmonic morphisms and totally geodesic horizontally conformal maps between compact Riemannian manifolds. In particular, it is shown that the only harmonic morphisms from a Riemannian symmetric space of compact type to a compact Riemann surface of genus at least 1 are the constant maps.