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This work considers the task of representation learning on the attributed relational graph (ARG). Both the nodes and edges in an ARG are associated with attributes/features allowing ARGs to encode rich structural information widely observed in real applications. Existing graph neural networks offer limited ability to capture complex interactions within local structural contexts, which hinders them from taking advantage of the expression power of ARGs. We propose Motif Convolution Module (MCM), a new motif-based graph representation learning technique to better utilize local structural information. The ability to handle continuous edge and node features is one of MCM's advantages over existing motif-based models. MCM builds a motif vocabulary in an unsupervised way and deploys a novel motif convolution operation to extract the local structural context of individual nodes, which is then used to learn higher-level node representations via multilayer perceptron and/or message passing in graph neural networks. When compared with other graph learning approaches to classifying synthetic graphs, our approach is substantially better in capturing structural context. We also demonstrate the performance and explainability advantages of our approach by applying it to several molecular benchmarks.