Ceramic matrix composites (CMCs) are a material of interest for components in the hot section of gas turbine engines due to their high strength-to-weight ratio and high temperature capabilities. CMCs are a class of material made of macro scale woven ceramic fibers infiltrated with a ceramic matrix making them significantly different than their nickel superalloy counterparts. As CMCs are implemented into gas turbine engines, the effects of the inherent topology of the CMC weave on convective heat transfer must be understood. In this study, film cooling holes were integrated into a representative CMC weave for three test coupons that were printed using additive manufacturing. The three coupons included: one having a weave topology along the surface of the internal channel supplying coolant to the film cooling holes; one having a weave topology along the external film cooled surface; and one having a weave topology on both the coolant supply channel as well as the external film cooled surface. Overall effectiveness levels for the two cases with a weave surface on the external film cooled wall were measured to be lower than levels for the case with the smooth external surface. The external weave significantly increased the mixing of the coolant jet with the hot mainstream resulting in poor cooling.