This work deals with the viscoelasticity of the arterial wall and its influence on the pulse waves. We describe the viscoelasticity by a nonlinear Kelvin–Voigt model in which the coefficients are fitted using experimental time series of pressure and radius measured on a sheep's arterial network. We obtained a good agreement between the results of the nonlinear Kelvin–Voigt model and the experimental measurements. We found that the viscoelastic relaxation time—defined by the ratio between the viscoelastic coefficient and the Young's modulus—is nearly constant throughout the network. Therefore, as it is well known that smaller arteries are stiffer, the viscoelastic coefficient rises when approaching the peripheral sites to compensate the rise of the Young's modulus, resulting in a higher damping effect. We incorporated the fitted viscoelastic coefficients in a nonlinear 1D fluid model to compute the pulse waves in the network. The damping effect of viscoelasticity on the high-frequency waves is clear especially at the peripheral sites.