A turbine airfoil includes: a concave pressure sidewall and a convex suction sidewall joined together at a leading edge and a trailing edge, and extending between a root and a tip; an internal rib extending between the pressure sidewall and the suction sidewall; and a crossover hole formed in the rib, the crossover hole having a noncircular cross-sectional shape with a major axis defining a maximum dimension of the cross-sectional shape; wherein the major axis of the crossover hole lies in plane with the rib and is non-parallel to an imaginary curvilinear lateral centerline which defines a locus of points lying halfway between the pressure and suction sidewalls. The orientation of the crossover holes minimizes stress concentration caused by the presence of the crossover holes.