The problem of structure-preserving model reduction of interconnected linear systems is considered in this paper. The problem is interesting because networked models often have high order, and standard model-reduction methods usually do not preserve interconnection structure. As a tool, balanced truncation and block-diagonal generalized controllability and observability Gramians are used. Block-diagonal generalized Gramians do not exist for all interconnected systems, but a class of systems that always has such Gramians is identified. Furthermore, it is shown how general interconnected systems can be associated with interconnected systems in this class. The block-diagonal Gramians are then used to compute the reduced models and also yield asymptotic a priori approximation error bounds and stability guarantees for the reduced models.