A method of direction-of-arrival (DoA) estimation for FMCW (Frequency Modulated Continuous Wave) radar is presented. In addition to MUSIC, which is the popular high-resolution DoA estimation algorithm, deep learning has recently emerged as a very promising alternative. It is proposed in this paper to use a 3D convolutional neural network (CNN) for DoA estimation. The 3D-CNN extracts from the radar data cube spectrum features of the region of interest (RoI) centered on the potential positions of the targets, thereby capturing the spectrum phase shift information, which corresponds to DoA, along the antenna axis. Finally, the results of simulations and experiments are provided to demonstrate the superior performance, as well as the limitations, of the proposed 3D-CNN.