Granular intrusion is common in a range of processes including ballistic impact and penetration problems, and locomotion of humans, animals, and vehicles in natural terrains. The computational cost of full-scale numerical modeling is high, whereas a capability to model such scenarios in real-time is critical in applications such as path planning and efficient maneuvering of vehicles in sandy terrains and extraterrestrial environments. Existing reduced-order methods that model intrusion have limited capabilities due to their shape-and media-specific forms. This work formulates a reduced-order modeling technique, 3-dimensional Resistive Force Theory (3D-RFT), capable of accurately and quickly predicting the resistive force on arbitrary-shaped bodies moving in grains. Aided by a continuum mechanical description of the granular bed, a comprehensive set of symmetry constraints, and a large amount of reference data, we develop a self-consistent and accurate form for 3D-RFT. We verify the model capabilities in a wide range of cases and show it can be quickly recalibrated to different media and intruder surface types.