Electrowetting-on-dielectric (EWOD) chips have become the most popular actuator particularly for droplet-based digital microfluidic (DMF) systems. In order to enable the electrical manipulations, wire routing is a key problem in designing EWOD chips. Unlike traditional very-large-scale-integration (VLSI) routing problems, in addition to routing-path establishment on signal pins, the EWOD-chip routing problem needs to address the issue of signal sharing for pin-count reduction under a practical constraint posed by limited pin-count supply. Moreover, EWOD-chip designs might incur several obstacles in the routing region due to embedded devices for specific fluidic protocols. However, no existing works consider the EWOD-chip routing with obstacles. To remedy this insufficiency, we propose in this paper the first obstacle-avoiding routing algorithm for pin-constrained EWOD chips. Our algorithm, based on effective integer-linear-programming (ILP) formulation as well as efficient routing framework, can achieve high routability with a low design complexity. Experimental results based on real-life chips with obstacles demonstrate the high routability of our obstacle-avoiding routing algorithm for pin-constrained EWOD chips.