Diffusion magnetic resonance imaging (dMRI) provides a unique tool for noninvasively probing the microstructure of the neuronal tissue. The NODDI model has been a popular approach to the estimation of tissue microstructure in many neuroscience studies. It represents the diffusion signals with three types of diffusion in tissue: intra-cellular, extra-cellular, and cerebrospinal fluid compartments. However, the original NODDI method uses a computationally expensive procedure to fit the model and could require a large number of diffusion gradients for accurate microstructure estimation, which may be impractical for clinical use. Therefore, efforts have been devoted to efficient and accurate NODDI microstructure estimation with a reduced number of diffusion gradients. In this work, we propose a deep network based approach to the NODDI microstructure estimation, which is named Microstructure Estimation using a Deep Network (MEDN). Motivated by the AMICO algorithm which accelerates the computation of NODDI parameters, we formulate the microstructure estimation problem in a dictionary-based framework. The proposed network comprises two cascaded stages. The first stage resembles the solution to a dictionary-based sparse reconstruction problem and the second stage computes the final microstructure using the output of the first stage. The weights in the two stages are jointly learned from training data, which is obtained from training dMRI scans with diffusion gradients that densely sample the q-space. The proposed method was applied to brain dMRI scans, where two shells each with 30 gradient directions (60 diffusion gradients in total) were used. Estimation accuracy with respect to the gold standard was measured and the results demonstrate that MEDN outperforms the competing algorithms.