The computational design of a 3D woven microvascular composite skin panel with straight and sinusoidal embedded microchannels for hypersonic aircraft applications is presented. To design this actively-cooled system, we determine the optimal values of the design parameters including the microchannels configuration, coolant flow rate, coolant type, and microchannels spacing to minimize the temperature, pressure drop in the microchannels, and the composite void volume fraction. A novel mesh-independent finite element method is implemented to evaluate the thermal response of the microvascular system. The Stream-line Upwind Petrov-Galerkin stabilization scheme is incorporated in the numerical solver to eliminate the spurious oscillations in the temperature field pertaining to the convective heat transfer in the microchannels. This study shows that for the given dimensions and applied thermal loads on the composite panel, the design of the system with straight microchannels yields the highest cooling efficiency. A design diagram is presented to determine an optimal set of the design parameters for a given maximum allowable temperature in the composite and the coolant, and a comparison is performed between parallel and counter-flow configurations.