A tip cooling apparatus for a turbine airfoil includes: a tip cap; a pair of spaced-apart tip walls connected to, extending around, and projecting outwardly from the tip cap so as to surround a central portion of the tip cap; a pocket defined by the tip walls; at least one feed hole passing through the tip cap or tip walls, communicating with the pocket; and a cooling matrix disposed in the pocket, the cooling matrix being an organized structure including an inlet surface having a plurality of inlets communicating with the pocket, and an outlet surface having a plurality of outlets, and further comprising a plurality of interior passages interconnecting the inlets to the outlets, with no line-of-sight therebetween.