This paper presents a theoretical investigation of the thermal characteristics of flat-micro heat pipes (FMHPs) with multi-heat sources and sinks. Analytical solutions of the pressure and the temperature distributions of FMHPs with multi-heat sources and sinks were obtained based on the modified liquid pressure drop. The solutions were used to identify the key engineering parameters of a mesh wick with microscale length that affect the maximum heat transfer rate of the FMHPs with multi-heat sources and sinks. The effects of the key engineering parameters on the maximum heat transfer rate of the FMHPs were presented for two limits. The first limit is the capillary limit and the other is the allowable maximum temperature limit which is used to ensure that the maximum surface temperature of the FMHP with the maximum heat transfer rate calculated at the capillary limit does not exceed the allowable maximum temperature of the electronic components. Finally, the theoretically results for the optimized wick structure for the corresponding maximum heat transfer rate and the surface temperature distribution of the FMHP were compared for the capillary limit only and for the maximum temperature limit cases, respectively.