Ori-kirigami structures offer a good avenue for designing mechanical metamaterials due to their unique advantage of being independent of material properties and scale limitations. Recently, the scientific community has been greatly interested in exploiting the complex energy landscape of ori-kirigami structures to construct multistable systems and play their valuable role in different applications. Here, we present three-dimensional ori-kirigami structures based on generalized waterbomb units, a cylindrical ori-kirigami structure based on waterbomb units, and a conical ori-kirigami structure based on trapezoidal waterbomb units. We investigate the inherent relationships between the unique kinematics and mechanical properties of these three-dimensional ori-kirigami structures and explore their potential usage as mechanical metamaterials that exhibit negative stiffness, snap-through, hysteresis effects, and multistability. What makes the structures even more attractive is their massive folding stroke, where the conical ori-kirigami structure can obtain a huge folding stroke of more than twice its initial height through penetration of its upper and lower boundaries. This study forms the foundation for designing and constructing three-dimensional ori-kirigami metamaterials based on generalized waterbomb units for various engineering applications.