Several hyperspectral (HS) systems based on compressive sensing (CS) theory have been presented to capture HS images with high accuracy and with a lower number of measurements than needed by conventional systems. However, the reconstruction of HS compressed measurements is time-consuming and commonly involves hyperparameter tuning per each scenario. In this paper, we introduce a Convolutional Neural Network (CNN) designed for the reconstruction of HS cubes captured with CS imagers based on spectral modulation. Our Deep Neural Network (DNN), dubbed DeepCubeNet, provides significant reduction in the reconstruction time compared to classical iterative methods. The performance of DeepCubeNet is investigated on simulated data, and we demonstrate for the first time, to the best of our knowledge, real reconstruction of CS HS measurements using DNN. We demonstrate significantly enhanced reconstruction accuracy compared to iterative CS reconstruction, as well as improvement in reconstruction time by many orders of magnitude.