We study the formation and propagation of nonlinear stress waves in 3D woodpile elastic metamaterials consisting of vertically stacked slender cylindrical rods. We find the nonlinear waveforms transmitted through the woodpile architectures under impact are highly sensitive to their design parameters, particularly stacking distances and angles in 3D assemblies. We numerically and experimentally demonstrate that the woodpile system can localize, modulate, and ultimately attenuate propagating nonlinear waves in an efficient manner without relying on material damping. We also show the feasibility of constructing metamaterials with a simultaneous characteristic of high damping and high stiffness by using the controllable wave dispersion in the woodpile structures. These woodpile metamaterials offer enhanced degrees of freedom in manipulating stress waves, thereby offering a new way to design efficient impact protectors with high stiffness.