Let (T M, G) and (T 1 M,G) respectively denote the tangent bundle and the unit tangent sphere bundle of a Riemannian manifold (M, g), equipped with arbitrary Riemannian g-natural metrics. After studying the geometry of the canonical projections π : (T M, G) → (M, g) and π 1 : (T 1 M,G) → (M, g), we give necessary and sufficient conditions for π and π 1 to be harmonic morphisms. Some relevant classes of Riemannian g-natural metrics will be characterized in terms of harmonicity properties of the canonical projections. Moreover, we study the harmonicity of the canonical projection : (T M − {0}, G) → (T 1 M,G) with respect to Riemannian g-natural metrics G,G of Kaluza-Klein type.