We give an algorithm to morph planar graph drawings that achieves small grid size at the expense of allowing a constant number of bends on each edge. The input is an n-vertex planar graph and two planar straight-line drawings of the graph on an O(n)×O(n) grid. The planarity-preserving morph is composed of O(n) linear morphs between successive pairs of drawings, each on an O(n)×O(n) grid with a constant number of bends per edge. The algorithm to compute the morph runs in O(n2) time on a word RAM model with standard arithmetic operations -- in particular no square roots or cube roots are required. The first step of the algorithm is to morph each input drawing to a planar orthogonal box drawing where vertices are represented by boxes and each edge is drawn as a horizontal or vertical segment. The second step is to morph between planar orthogonal box drawings. This is done by extending known techniques for morphing planar orthogonal drawings with point vertices.