A myriad of nuclear reactor designs are currently being considered for next-generation power production. These designs utilize unique geometries and materials and can rely on multiphysics effects for safety and operation. This work develops a neutron transport tool, MOCkingbird, capable of three-dimensional (3D), full-core reactor simulation for previously intractable geometries. The solver is based on the method of characteristics, utilizing unstructured mesh for the geometric description. MOCkingbird is built using the MOOSE multiphysics framework, allowing for straightforward linking to other physics in the future. A description of the algorithms and implementation is given, and solutions are computed for two-dimensional/3D C5G7 and the Massachusetts Institute of Technology BEAVRS benchmark. The final result shows the application of MOCkingbird to a 3D, full-core simulation utilizing 1.4 billion elements and solved using 12 000 processors.