Let $φ\colon Γ_2 \rightarrow Γ_1$ be a harmonic morphism of connected graphs. We show that an arithmetical structure on $Γ_1$ can be pulled back via $φ$ to an arithmetical structure on $Γ_2$. We then show that some results of Baker and Norine on the critical groups for the usual Laplacian extend to arithmetical critical groups, which are abelian groups determined by the generalized Laplacian associated to these arithmetical structures. In particular, we show that the morphism $φ$ induces a surjective group homomorphism from the arithmetical critical group of $Γ_2$ to that of $Γ_1$ and an injective group homomorphism from the arithmetical critical group of $Γ_1$ to that of $Γ_2$. Finally, we prove a Riemann-Hurwitz formula for arithmetical structures.