An axial-flow blood pump has a rotor suspended in ball-and-cup bearings which are blood-cooled but not actively blood-lubricated. The ball-and-cup structures are made of highly heat-conductive material and are in heat-transferring contact with heat-conductive stator blades that serve as heat sinks for the bearings. The ball-and-cup structures are radially much smaller than the stator blades. The ball-and-cup interface has so small a gap that the ball-to-cup structures present an essentially continuous surface to the blood flow.