Syntactic information, especially dependency trees, has been widely used by existing studies to improve relation extraction with better semantic guidance for analyzing the context information associated with the given entities. However, most existing studies suffer from the noise in the dependency trees, especially when they are automatically generated, so that intensively leveraging dependency information may introduce confusions to relation classification and necessary pruning is of great importance in this task. In this paper, we propose a dependency-driven approach for relation extraction with attentive graph convolutional networks (A-GCN). In this approach, an attention mechanism upon graph convolutional networks is applied to different contextual words in the dependency tree obtained from an off-the-shelf dependency parser, to distinguish the importance of different word dependencies. Consider that dependency types among words also contain important contextual guidance, which is potentially helpful for relation extraction, we also include the type information in A-GCN modeling. Experimental results on two English benchmark datasets demonstrate the effectiveness of our A-GCN, which outperforms previous studies and achieves state-of-the-art performance on both datasets.