3D motion estimation from cine cardiac magnetic resonance (CMR) images is important for the assessment of cardiac function and the diagnosis of cardiovascular diseases. Current state-of-the art methods focus on estimating dense pixel-/voxel-wise motion fields in image space, which ignores the fact that motion estimation is only relevant and useful within the anatomical objects of interest, e.g., the heart. In this work, we model the heart as a 3D mesh consisting of epi- and endocardial surfaces. We propose a novel learning framework, DeepMesh, which propagates a template heart mesh to a subject space and estimates the 3D motion of the heart mesh from CMR images for individual subjects. In DeepMesh, the heart mesh of the end-diastolic frame of an individual subject is first reconstructed from the template mesh. Mesh-based 3D motion fields with respect to the end-diastolic frame are then estimated from 2D short- and long-axis CMR images. By developing a differentiable mesh-to-image rasterizer, DeepMesh is able to leverage 2D shape information from multiple anatomical views for 3D mesh reconstruction and mesh motion estimation. The proposed method estimates vertex-wise displacement and thus maintains vertex correspondences between time frames, which is important for the quantitative assessment of cardiac function across different subjects and populations. We evaluate DeepMesh on CMR images acquired from the UK Biobank. We focus on 3D motion estimation of the left ventricle in this work. Experimental results show that the proposed method quantitatively and qualitatively outperforms other image-based and mesh-based cardiac motion tracking methods.