A cooling system for use in a turbine engine component exposed to high temperatures during engine operation. The system includes a serpentine flow passage and an exhaust region. The serpentine flow passage includes a coolant supply inlet. The passage can be configured so that neighboring portions of the passage have coolant flowing in the same direction or, alternatively, in opposite directions. A number of flow disrupting structures, such as microfins and trip strips, can be located along the flow passage. The exhaust region can discharge coolant from the system at reduced exit momentum. The exiting flow can provide film cooling to the component. The cooling system can be provided in a small modular form, which can increase cooling design flexibility and can allow cooling designs tailored to the unique cooling requirements of the individual component. As a result, the modules can result in high levels of cooling effectiveness.