In this work, a microfluidic system to investigate the flow behavior of red blood cells in a microcirculation-mimicking network of PDMS microchannels with thickness comparable to cell size is presented. We provide the first quantitative description of cell velocity and shape as a function of the applied pressure drop in such devices. Based on these results, a novel methodology to measure cell membrane viscoelastic properties in converging/diverging flow is developed, and the results are in good agreement with data from the literature. In particular, in the diverging channel the effect of RBC surface viscosity is dominant with respect to shear elasticity. Possible applications include measurements of cell deformability in pathological samples, where reliable methods are still lacking.