Photoluminescence (PL) of organometal halide perovskite materials reflects the charge dynamics inside of the material and thus contains important information for understanding the electro-optical properties of the material. Interpretation of PL blinking of methylammonium lead iodide (MAPbI3) nanostructures observed on polycrystalline samples remains puzzling owing to their intrinsic disordered nature. Here, we report a novel method for the synthesis of high-quality single-crystal MAPbI3 nanorods and demonstrate a single-crystal study on MAPbI3 PL blinking. At low excitation power densities, two-state blinking was found on individual nanorods with dimensions of several hundred nanometers. A super-resolution localization study on the blinking of individual nanorods showed that single crystals of several hundred nanometers emit and blink as a whole, without showing changes in the localization center over the crystal. Moreover, both the blinking ON and OFF times showed power-law distributions, indicating trapping-detrapping processes. This is further supported by the PL decay times of the individual nanorods, which were found to correlate with the ON/OFF states. Furthermore, a strong environmental dependence of the nanorod PL blinking was revealed by comparing the measurements in vacuum, nitrogen, and air, implying that traps locate close to crystal surfaces. We explain our observations by proposing surface charge traps that are likely related to under-coordinated lead ions and methylammonium vacancies to result in the PL blinking observed here.