Semantic parses are directed acyclic graphs (DAGs), so semantic parsing should be modeled as graph prediction. But predicting graphs presents difficult technical challenges, so it is simpler and more common to predict the linearized graphs found in semantic parsing datasets using well-understood sequence models. The cost of this simplicity is that the predicted strings may not be well-formed graphs. We present recurrent neural network DAG grammars, a graph-aware sequence model that ensures only well-formed graphs while sidestepping many difficulties in graph prediction. We test our model on the Parallel Meaning Bank---a multilingual semantic graphbank. Our approach yields competitive results in English and establishes the first results for German, Italian and Dutch.