Data-to-text (D2T) generation aims to transform structured data into natural language text. Data-to-text pre-training has proved to be powerful in enhancing D2T generation and yields impressive performance. However, previous pre-training methods either oversimplified structured data into a sequence without considering input structures or designed training objectives tailored for a specific data structure (e.g., table or knowledge graph). In this paper, we unify different types of structured data (i.e., table, key-value data, knowledge graph) into the graph format and cast different D2T generation tasks as graph-to-text generation. To effectively exploit the structural information of the input graph, we propose a structure-enhanced pre-training method for D2T generation by designing a structure-enhanced Transformer. Concretely, we devise a position matrix for the Transformer, encoding relative positional information of connected nodes in the input graph. In addition, we propose a new attention matrix to incorporate graph structures into the original Transformer by taking the available explicit connectivity structure into account. Extensive experiments on six benchmark datasets show the effectiveness of our model. Our source codes are available at https://github.com/AlibabaResearch/DAMO-ConvAI/tree/main/unid2t.