Convolutional neural networks (CNNs) leverage the great power in representation learning on regular grid data such as image and video. Recently, increasing attention has been paid on generalizing CNNs to graph or network data which is highly irregular. Some focus on graph-level representation learning while others aim to learn node-level representations. These methods have been shown to boost the performance of many graph-level tasks such as graph classification and node-level tasks such as node classification. Most of these methods have been designed for single-dimensional graphs where a pair of nodes can only be connected by one type of relation. However, many real-world graphs have multiple types of relations and they can be naturally modeled as multi-dimensional graphs with each type of relation as a dimension. Multi-dimensional graphs bring about richer interactions between dimensions, which poses tremendous challenges to the graph convolutional neural networks designed for single-dimensional graphs. In this paper, we study the problem of graph convolutional networks for multi-dimensional graphs and propose a multi-dimensional convolutional neural network model mGCN aiming to capture rich information in learning node-level representations for multi-dimensional graphs. Comprehensive experiments on real-world multi-dimensional graphs demonstrate the effectiveness of the proposed framework.