Three-dimensional structured illumination microscopy (3D-SIM) plays an essential role in biological volumetric imaging with the capabilities of improving lateral and axial resolution. However, the traditional linear 3D algorithm is sensitive to noise and generates artifacts, while the low temporal resolution hinders live-cell imaging. In this paper, we propose a novel 3D-SIM algorithm based on total variation (TV) and fast iterative shrinkage threshold algorithm (FISTA), termed TV-FISTA-SIM. Compared to conventional algorithms, TV-FISTA-SIM achieves higher reconstruction fidelity with the least artifacts, even when the signal-to-noise ratio (SNR) is as low as 5 dB, and a faster reconstruction rate. Through simulation, we have verified that TV-FISTA-SIM can effectively reduce the amount of required data with less deterioration. Moreover, we demonstrate TV-FISTA-SIM for high-quality multi-color 3D super-resolution imaging, which can be potentially applied to live-cell imaging applications.