A turbine airfoil (60) includes an airfoil outer wall (66) that defines leading and trailing ends (LE,TE) and first and second sides (68a,68b) that join the leading and trailing ends (LE,TE). At least one cooling passage (74) is embedded in the airfoil outer wall (66) and has a radially-elongated entrance manifold (76), a radially-elongated diffuser orifice (78) that opens to an exterior surface of the airfoil outer wall (66), and a bank of sub-passages (80) fluidly connecting the radially-elongated entrance manifold (76) with the radially-elongated diffuser orifice (78). The radially-elongated diffuser orifice (78) is sloped relative to the radially-elongated entrance manifold (76).