The large eddy simulation (LES) of the flow around a circular cylinder is not only affected by the sub-grid scale (SGS) model but also by the grid resolution of the computational domain. To study the influence of different grids on the LES results, the LES simulations of the flow around a circular cylinder with different grids at Reynolds number (Re) = 3900 was performed. A circular computational domain with different radial growth rates and circumferential and spanwise grid numbers was adopted for the simulations. Meanwhile, the aerodynamic forces, wind pressure coefficients, mean and instantaneous flow fields, and the effect of grid resolution on them were comprehensively analyzed. The results indicate that the lift coefficient, wind pressure coefficient, and recirculation length are significantly affected by the radial growth rate of the grid and the circumferential grid number. The spanwise grid number has a significant influence on the three-dimensionality of the flow and plays an important role in velocity fluctuations in the wake region. Nevertheless, the aerodynamic coefficients and recirculation length are not sufficiently sensitive to the grid number in the spanwise direction. By comparing the results, it can be concluded that suitable and reliable LES results can be obtained when the radial growth rate is 1.03 or 1.05, the circumferential grid number is 160, 200, or 240, and the spanwise grid number is 64. A radial growth rate 1.05, circumferential grid number 160, and spanwise grid number 64 are recommended to reduce the grid amount and further improve the efficiency.