Accurate segmentation of kidneys and kidney tumors is an essential step for radiomic analysis as well as developing advanced surgical planning techniques. In clinical analysis, the segmentation is currently performed by clinicians from the visual inspection of images gathered through a computed tomography (CT) scan. This process is laborious and its success significantly depends on previous experience. We present a multi-scale supervised 3D U-Net, MSS U-Net to segment kidneys and kidney tumors from CT images. Our architecture combines deep supervision with exponential logarithmic loss to increase the 3D U-Net training efficiency. Furthermore, we introduce a connected-component based post processing method to enhance the performance of the overall process. This architecture shows superior performance compared to state-of-the-art works, with the Dice coefficient of kidney and tumor up to 0.969 and 0.805 respectively. We tested MSS U-Net in the KiTS19 challenge with its corresponding dataset.