Higher-rank graphs (k-graphs) are a combinatorial model for C*-algebras; indeed, much of the structure of k-graph C*-algebras is encoded in the underlying combinatorial data of the k-graph. However, different k-graphs can give rise to isomorphic or Morita equivalent C*-algebras. In this talk, we present several ways to modify the structure of a k-graph which preserve the Morita equivalence class of the associated C*-algebra. Our constructions are inspired by the analogous work for graph C*-algebras of Bates and Pask, as well as by the textile system approach to describing k-graphs. This is joint work with C. Eckhardt, K. Fieldhouse, D. Gent, I. Gonzales, and D. Pask.