Ceramic matrix composites (CMCs) are of interest for hot section components of gas turbine engines due to their low weight and favorable thermal properties. To implement this advanced composite in a gas turbine engine, characterizing the influence of CMC's surface topology on heat transfer and cooling performance is critical. However, very few published studies have reported the flow and heat transfer effects caused by this unique surface topology. This study is an experimental and computational investigation to evaluate the effect of weave orientations, relevant to CMC surfaces, on the resulting pressure loss and convective heat transfer within an internal channel. The weave pattern was additively manufactured as the walls of a scaled-up coupon containing a single channel. For each of the three weave orientations, bulk pressure losses and convective heat transfer coefficients were measured over a range of Reynolds numbers. Scaling the pressure losses in terms of a friction factor and convective heat transfer coefficients in terms of a Nusselt number showed the importance of choosing the appropriate definition of the hydraulic diameter, which was particularly important for the friction factor. A coupon having one wall with the weave surface increased pressure loss and heat transfer compared to a smooth wall with the largest increases occurring when the CMC weave strands were perpendicular to the flow. Friction factor augmentations were much higher than heat transfer augmentations. When adding the weave to a second channel wall, pressure loss and heat transfer were further increased. Orienting the CMC strands perpendicular to the flow consistently showed the largest augmentations in heat transfer over a smooth channel, but at a much higher pressure loss penalty than that seen with the CMC strands parallel to the flow.