A theorem of Leighton says that two finite graphs with isomorphic universal covers have a common finite cover. In group theoretic terms, Leighton's theorem says that any two uniform lattices in the automorphism group of a tree are weakly commensurable. We introduce the class of graphically discrete groups, a large class of groups that includes nilpotent groups, most lattices in semisimple Lie groups, mapping class groups and fundamental groups of closed connected irreducible 3-manifolds. We then prove a Leighton-style rigidity theorem for free products of torsion-free graphically discrete groups, showing that if two such groups are uniform lattices in the same locally compact group, then they are abstractly commensurable. This result applies to fundamental groups of closed connected non-prime 3-manifolds. Joint with Sam Shepherd, Emily Stark and Daniel Woodhouse.