Evaluation of dynamic soil properties under shear and compressional waves is an important concern in geotechnical earthquake engineering. In this work, the nonlinear strain-dependent behaviors of stiffness and damping properties of granular materials with different particle size distributions are studied using three-dimensional discrete element method. In particular, both nonlinear variations in shear and constrained modulus and associated damping ratios with strain are analyzed based on the simulations of cyclic triaxial and cyclic constrained compression tests. Micromechanical explorations on the influence of coefficient of uniformity (C_u) and the manner of cyclic loading are presented. The results indicate that the shear modulus exhibits a reduction relationship with shear strain in both triaxial compression and extension stages, whereas the constrained modulus only degrades during the extension stage. The damping ratio under cyclic triaxial loading is much larger than that under cyclic constrained compression. The dynamic nonlinearity under two types of cyclic loading is more pronounced with increasing C_u. The more significant modulus reduction with higher C_u correlates with a lower percentage of strong contact and a more heterogeneous distribution of strong contact normal forces. The increase in damping ratio with higher C_u is associated with a larger percentage of sliding contacts. The difference in damping ratio between cyclic triaxial and cyclic constrained compression tests is attributed to the different strain energy magnitudes that are required to the target shear and compression strain levels.