Heat transfer enhancement for a steady, laminar, incompressible and axially fully developed flow of nonlinear viscoelastic shear-thinning fluids abiding by the Giesekus constitutive structure in straight microtubes of arbitrary non-circular contour under constant heat flux at the walls is investigated. The governing equations are solved based on an embedded double asymptotic series expansion pivoted around the elasticity measure Weissenberg number Wi and a mapping parameter e. The effects of the constitutive parameters of the Giesekus model as well as the dimensionless flow parameters on the flow and heat transfer in microtubes of a wide-ranging spectrum of non-circular cross-sectional contours are investigated. The flow and thermal fields of representative tubes with equilateral triangular, square, rectangular, and pentagonal cross-sections are investigated in detail. The analysis reveals that the constitutive mobility parameter alpha, viscosity ratio /3 and the Weissenberg number Wi significantly influence the Nusselt number Nu together with the flow characteristics, the Pe<acute accent>clet Pe, Brinkman Br and Reynolds Re numbers. Effects of the viscous dissipation in wall cooling and wall heating is studied. At a given value of Wi heat transfer is enhanced regardless the process considered at the boundary. In all cases, heat transfer is improved compared to Newtonian fluids, with the strongest enhancement observed in rectangular microtubes for low-inertia flows.