Embodiments of the invention relate to various cooling systems for a turbine vane made of stacked ceramic matrix composite (CMC) laminates. Each airfoil-shaped laminate has an in-plane direction and a through thickness direction substantially normal to the in-plane direction. The laminates have anisotropic strength characteristics in which the in-plane tensile strength is substantially greater than the through thickness tensile strength. Such a vane construction lends itself to the inclusion of various cooling features in individual laminates using conventional manufacturing and forming techniques. When assembled in a radial stack, the cooling features in the individual laminates can cooperate to form intricate three dimensional cooling systems in the vane.