Micro heat exchangers may achieve very high heat transfer coefficients thanks to their small dimensions and high area-to-volume ratio even in laminar flow, a highly desirable feature in many industrial applications. The main drawback of these devices is the high frictional lossesespecially for liquid flowsthat make viscous dissipation no longer negligible. In order to enhance heat transfer, modification of the channels' cross-section is a viable strategy. In this work the fully developed steady laminar flow of a Newtonian liquid through a microchannel subject to uniform heat flux, uniform peripheral temperature boundary conditions in the presence of viscous dissipation is investigated. Entropy generation numbers and the constrained total heat transfer area performance evaluation criterion are employed to assess the influence of smoothing the corners of an initially rectangular cross section, with an aspect ratio ranging from 1 to 0.03 under four different types of geometrical constraints. The governing equations and the results are expressed in nondimensional form, with the intensity of viscous dissipation being exemplified by the Brinkman number, which is demonstrated to increase its maximum allowable value when corners are smoothed. The results are reported as a function of the nondimensional radius of curvature and aspect ratio and show that smoothing the corners almost invariably brings an improvement for a fixed heated perimeter.