A catheter using multiple hollow fibers to carry a heating/cooling fluid employs fiber spreading features to improve heat transfer and help prevent clotting of body fluids within which the catheter is immersed. The catheter includes a length of outer tube surrounding an inner tube. The inner tube conveys fluids in one direction, and the passageway between inner and outer tubes conveys fluids in the opposite direction. The inner and outer tubes have proximal distal ends. The distal end is closed, but the proximal end is open to permit introduction of a heating or cooling fluid into the inner tube, and withdrawal of the fluid from the passageway between the inner and outer tubes. At the tubes' distal end resides a proximal fluid transfer housing coupled to multiple hollow heat exchange fibers. The distal ends of these fibers may be commonly connected to a distal fluid transfer housing and optional reservoir, which cooperatively return fluid to the passageway between inner and outer tubes. In another embodiment, each fiber includes both outflowing and return lumens, enabling the fibers' distal ends to freely move about. In another embodiment, each fiber proceeds outward from the proximal fluid transfer housing, and returns back to form a loop. In each different embodiment, the invention includes one or more actuating structures to spread the fibers. The actuating structure may include, for example, fiber shaping, a retractable spreading member, elastic bias built into the fibers, etc.