Blood pulsatility, aneurysms, stenoses and general low shear stress hemodynamics enhance non-Newtonian blood effects which generate local changes in the space-time evolution of the blood pressure, flow rate and cross-sectional area of elastic vessels. Even though these local changes are known to cause global unexpected hemodynamical behaviors, all one-dimensional (1D) blood flow models are built under Newtonian fluid hypothesis. In this work, we present a time-dependent non-Newtonian extension of a 1D blood flow model, able to describe local space-time variations of the viscous behavior of blood. The rheological model is based on a simplified Maxwell viscoelastic equation for the shear stress with structure dependent coefficients. We compare the numerical predictions of the 1D non-Newtonian model to experimental rheological data available in the literature. Specifically, we explore four well documented shear stress protocols and we show that the results predicted by the 1D non-Newtonian model in a single artery accurately compare, both qualitatively and quantitatively, to the steady and unsteady shear stresses measured experimentally. We then use the 1D non-model to compute the flow in idealized healthy and pathological symmetric and asymmetric networks of increasing size. We show that aggregation occurs in such networks occurs, leading to non-Newtonian blood behaviors especially in the presence of pathologies. This non-Newtonian extension of a 1121 blood flow model will be useful in the future to improve our understanding of the large-scale hemodynamics in micro- and macro-circulation networks.