In one aspect the present subject matter is directed to a system for cooling a turbine shroud. The system includes a cooling medium source that provides a cooling medium, and a turbine shroud support assembly that includes a shroud support ring. A shroud seal is coupled to the turbine shroud support assembly. The shroud seal includes a back side surface and an inner surface. A pin extends at least partially through the shroud support ring towards the back side surface. The pin includes a flow passage having an inlet that is in fluid communication with the cooling medium source and an aperture that is disposed downstream from the inlet. The aperture is oriented to direct a flow of the cooling medium out of the flow passage in a flow direction that is non-perpendicular to the back side surface of the shroud seal.