3D Multi-object tracking (MOT) is crucial to autonomous systems. Recent work often uses a tracking-by-detection pipeline, where the feature of each object is extracted independently to compute an affinity matrix. Then, the affinity matrix is passed to the Hungarian algorithm for data association. A key process of this pipeline is to learn discriminative features for different objects in order to reduce confusion during data association. To that end, we propose two innovative techniques: (1) instead of obtaining the features for each object independently , we propose a novel feature interaction mechanism by introducing Graph Neural Networks; (2) instead of obtaining the features from either 2D or 3D space as in prior work, we propose a novel joint feature extractor to learn appearance and motion features from 2D and 3D space. Through experiments on the KITTI dataset, our proposed method achieves state-of-the-art 3D MOT performance. Our project website is at http://www.xinshuoweng.com/projects/GNN3DMOT.