Due to the cost and complexity of biological experiments, many computational methods have been proposed to predict potential miRNA-disease associations by utilizing known miRNA-disease associations and other related information. However, there are some challenges for these computational methods. First, the relationships between miRNAs and diseases are complex. The computational network should consider the local and global influence of neighborhoods from the network. Furthermore, predicting disease-related miRNAs without any known associations is also very important. This study presents a new computational method that constructs a heterogeneous network composed of a miRNA similarity network, disease similarity network, and known miRNA-disease association network. The miRNA similarity considers the miRNAs and their possible families and clusters. The information of each node in heterogeneous network is obtained by aggregating neighborhood information with graph convolutional networks (GCNs), which can pass the information of a node to its intermediate and distant neighbors. Disease-related miRNAs with no known associations can be predicted with the reconstructed heterogeneous matrix. We apply 5-fold cross-validation, leave-one-disease-out cross-validation, and global and local leave-one-out cross-validation to evaluate our method. The corresponding areas under the curves (AUCs) are 0.9616, 0.9946, 0.9656, and 0.9532, confirming that our approach significantly outperforms the state-of-the-art methods. Case studies show that this approach can effectively predict new diseases without any known miRNAs.