A three-dimensional model is developed to investigate how and conjugate heat transfer in the microchannel-based heat sink for electronic packaging applications. The incompressible laminar Navier-Stokes equations of motion are employed as the governing conservation equations which are numerically solved using the generalized single-equation framework for solving conjugate problems. The theoretical model developed is validated by comparing the predictions of the thermal resistance and the friction coefficient with available experimental data for a wide range of Reynolds numbers. The detailed temperature and heat flux distributions as well as the average heat transfer characteristics are reported and discussed. The analysis provides a unique fundamental insight into the complex heat flow pattern established in the channel due to combined convection-conduction effects in the three-dimensional setting. Important practical recommendations are also provided regarding the cooling efficiency of the microchannel heat sinks as well as a possible failure due to the thermal stresses induced by the extremely large temperature gradient at the entrance of the channels.