The shift towards multi-core architectures has ushered in a new era of shared memory parallelism for scientific applications. This transition has introduced challenges for the nuclear engineering community as it seeks to design high-fidelity full-core reactor physics simulation tools. This paper describes the parallel transport sweep algorithm in the OpenMOC method of characteristics (MOC) neutron transport code for multi-core platforms using OpenMP. Strong and weak scaling studies are performed for both Intel Xeon and IBM Blue Gene/Q multi-core processors. The results demonstrate 100% parallel efficiency for 12 threads on 12 cores on Intel Xeon platforms, and over 90% parallel efficiency with 64 threads on 16 cores on the IBM Blue Gene/Q. These results illustrate the potential for hardware acceleration for MOC neutron transport on modern multi-core and future many-core architectures. In addition, this work highlights the pitfalls of programming for multi-core architectures, with a focal point on false sharing.