Machine learning (ML) is becoming increasingly popular for predicting material properties to accelerate materials discovery. Because material properties are strongly affected by its crystal structure, a key issue is converting the crystal structure into the features for input to the ML model. Currently, the most common method is to convert the crystal structure into a graph and predicting its properties using a graph neural network (GNN). Some GNN models, such as crystal graph convolutional neural network (CGCNN) and atomistic line graph neural network (ALIGNN), have achieved highly accurate predictions of material properties. Despite these successes, using a graph to represent a crystal structure has the notable limitation of losing the crystal structure's three-dimensional (3D) information. In this work, we propose DeepCrysTet, a novel deep learning approach for predicting material properties, which uses crystal structures represented as a 3D tetrahedral mesh generated by Delaunay tetrahedralization. DeepCrysTet provides a useful framework that includes a 3D mesh generation method, mesh-based feature design, and neural network design. The experimental results using the Materials Project dataset show that DeepCrysTet significantly outperforms existing GNN models in classifying crystal structures and achieves state-of-the-art performance in predicting elastic properties.