We consider the effect of modulating the microchannel shapes within a liquid-cooled device to determine how geometry affects heat transfer effects. By using the asymptotic approach of homogenization, we find a single advection-diffusion equation, which has an anisotropic thermal conductivity tensor that depends on the local channel width and laminate thickness. A novel modification of the homogenization approach is used to allow the spacing of the microchannels and laminates to vary over the macroscale, relaxing the spatial periodicity requirement of the technique. We find that the anisotropic thermal conductivity corresponds to two different classes of steady thermal solutions: a hot spot symmetric along the centerline, or two maxima which are symmetric about the centerline. The location of the hot spot depends on the energy balance between convective transport in the streamwise direction and effective conduction in the spanwise direction. Optimiziation formulations for families of channel/laminate geometries are performed to find minimum average temperatures within each of these families. (C) 2013 Elsevier Ltd. All rights reserved.