In this paper we study the vanishing viscosity limit of strictly hyperbolic systems, extending the earlier result in [A. Bressan and T. Yang, Comm. Pure Appl. Math., 57 (2004), pp. 1075--1109] to systems where each characteristic field can be either genuinely nonlinear or linearly degenerate. For a given initial data with small total variation, our main estimate shows that the ${\bf L}^1$ distance between the exact solution $u$ and a viscous approximation $u^\varepsilon $ is bounded by $\|u(\tau,\cdot)-u^\varepsilon (\tau,\cdot)\|_{{\bf L}^1} = O(1)\cdot(1+\tau)\varepsilon^{1/4}.$ Under the additional assumptions that the integral curves of all linearly degenerate fields are straight lines, we obtain the sharper estimate $\|u(\tau,\cdot)-u^\varepsilon (\tau,\cdot)\|_{{\bf L}^1} = O(1)\cdot(1+\tau)\sqrt\varepsilon |\ln\varepsilon |.$