Los Alamos National Laboratory Monte Carlo N-Particle transport code (MCNP) Version 6 (MCNP6) has been extended to include a new capability that permits tracking of neutrons and photons on an unstructured mesh that is embedded as a mesh universe within its constructive solid geometry capability. The mesh geometry is created through Abaqus/CAE using its solid modeling capabilities. MCNP transport results are calculated for mesh elements using a path length estimator while element-to-element tracking is performed on the mesh. The results from MCNP6 can be exported to Abaqus/CAE for visualization or other physics analysis. Three geometrically simple benchmark experiments were analyzed: Godiva, Osaka nickel sphere, and fusion neutron source vanadium cube. Computer run time is proportional to the number of mesh elements, element order, and element type specified in the input. Good agreement of our MCNP6 results with the measured neutron leakage for the nickel sphere and the measured neutron and gamma spectra from the vanadium assembly was observed.