Run, run, as fast as you can. Also, be careful, as lag can cause you to fall through the bridge to your death.
Take turns running across the bridge to the cheese and back. In a room with enough people, the bridge may break if all the mice rush across at once. If the bridge seems like it will be broken, stay at one of the two ends and wait until it looks safe or until the Shaman creates a new path over the abyss.
Try and support the bridge using red-anchored planks to increase the amount of mice that can safely travel on it at any one time. Should the bridge fail, which it often does, construct a new bridge across the gap. Do not put an anvil or use a cannonball directed at the bridge, because it will break. Troll Shamans will often do this.
Run to the middle and immediately place a globally anchored plank under the bridge. If the mice are willing to wait and with enough time, adding rotating arrows to every joint between the bridge plank will cause it to "tense" curve upwards, and become nearly unbreakable even with a cannon.
Another very common solution is to fill the lower part of the bridge with balloons, which counter the weight of mice and even anvils.