Text-to-SQL aims to generate an executable SQL program given the user utterance and the corresponding database schema. To ensure the well-formedness of output SQLs, one prominent approach adopts a grammar-based recurrent decoder to produce the equivalent SQL abstract syntax tree (AST). However, previous methods mainly utilize an RNN-series decoder, which 1) is time-consuming and inefficient and 2) introduces very few structure priors. In this work, we propose an AST structure-aware Transformer decoder (ASTormer) to replace traditional RNN cells. The structural knowledge, such as node types and positions in the tree, is seamlessly incorporated into the decoder via both absolute and relative position embeddings. Besides, the proposed framework is compatible with different traversing orders even considering adaptive node selection. Extensive experiments on five text-to-SQL benchmarks demonstrate the effectiveness and efficiency of our structured decoder compared to competitive baselines.