With reducing transistor sizes and increasing levels of integration, extracting the heat from electronic devices is an ongoing challenge. Conventional indirect-cooling approaches are hindered by thermal interfaces, as well as heat spreading to larger areas which prevents dense integration. This work presents cooling strategies with microchannels directly embedded inside GaN-based semiconductor dies to extract higher heat fluxes, as well as substantially reduce pumping power. An experimental comparison is made between three levels of microchannel cooling: Indirect, using a thermal interface; direct embedded in the backside of the die; and co-designed, where the cooling channels are fundamentally integrated in the design of the electronic device. We show that, by having a cooling-centered device design, heat fluxes exceeding 1.75 kW/cm2 can be extracted with very high efficiency. Finally, to complement the compact cooling system, we discuss approaches for coolant delivery, and show how a new PCB-based coolant distribution can be used to obtain very-compact power converters, that may support the electrification of our society in the future.