The noise in diffusion-weighted images (DWIs) decreases the accuracy and precision of diffusion tensor magnetic resonance imaging (DTI) derived microstructural parameters and leads to prolonged acquisition time for achieving improved signal-to-noise ratio (SNR). Deep learning-based image denoising using convolutional neural networks (CNNs) has superior performance but often requires additional high-SNR data for supervising the training of CNNs, which reduces the practical feasibility. We develop a self-supervised deep learning-based method entitled "SDnDTI" for denoising DTI data, which does not require additional high-SNR data for training. Specifically, SDnDTI divides multi-directional DTI data into many subsets, each consisting of six DWI volumes along optimally chosen diffusion-encoding directions that are robust to noise for the tensor fitting, and then synthesizes DWI volumes along all acquired directions from the diffusion tensors fitted using each subset of the data as the input data of CNNs. On the other hand, SDnDTI synthesizes DWI volumes along acquired diffusion-encoding directions with higher SNR from the diffusion tensors fitted using all acquired data as the training target. SDnDTI removes noise from each subset of synthesized DWI volumes using a deep 3-dimensional CNN to match the quality of the cleaner target DWI volumes and achieves even higher SNR by averaging all subsets of denoised data. The denoising efficacy of SDnDTI is demonstrated on two datasets provided by the Human Connectome Project (HCP) and the Lifespan HCP in Aging. The SDnDTI results preserve image sharpness and textural details and substantially improve upon those from the raw data. The results of SDnDTI are comparable to those from supervised learning-based denoising and outperform those from state-of-the-art conventional denoising algorithms including BM4D, AONLM and MPPCA.