Machine translation has seen rapid progress with the advent of Transformer-based models. These models have no explicit linguistic structure built into them, yet they may still implicitly learn structured relationships by attending to relevant tokens. We hypothesize that this structural learning could be made more robust by explicitly endowing Transformers with a structural bias, and we investigate two methods for building in such a bias. One method, the TP-Transformer, augments the traditional Transformer architecture to include an additional component to represent structure. The second method imbues structure at the data level by segmenting the data with morphological tokenization. We test these methods on translating from English into morphologically rich languages, Turkish and Inuktitut, and consider both automatic metrics and human evaluations. We find that each of these two approaches allows the network to achieve better performance, but this improvement is dependent on the size of the dataset. In sum, structural encoding methods make Transformers more sample-efficient, enabling them to perform better from smaller amounts of data.