We study harmonic Riemannian maps on locally conformal Kaehler manifolds (IcK manifolds). We show that if a Riemannian holomorphic map between IcK manifolds is harmonic, then the Lee vector field of the domain belongs to the kernel of the Riemannian map under a condition. When the domain is Kaehler, we prove that a Riemannian holomorphic map is harmonic if and only if the IcK manifold is Kaehler. Then we find similar results for Riemannian maps between IcK manifolds and Sasakian manifolds. Finally, We check the constancy of some maps between almost complex (or almost contact) manifolds and almost product manifolds.