Multistep traffic forecasting on road networks is a crucial task in successful intelligent transportation system applications. To capture the complex non-stationary temporal dynamics and spatial dependency in multistep traffic-condition prediction, we propose a novel deep learning framework named attention graph convolutional sequence-to-sequence model (AGC-Seq2Seq). In the proposed deep learning framework, spatial and temporal dependencies are modeled through the Seq2Seq model and graph convolution network separately, and the attention mechanism along with a newly designed training method based on the Seq2Seq architecture is proposed to overcome the difficulty in multistep prediction and further capture the temporal heterogeneity of traffic pattern. We conduct numerical tests to compare AGC-Seq2Seq with other benchmark models using a real-world dataset. The results indicate that our model yields the best prediction performance in terms of various prediction error measures. Furthermore, the variation of spatiotemporal correlation of traffic conditions under different perdition steps and road segments is revealed through sensitivity analyses.