Before the Han Dynasty, what were the main media of communication used by the Chinese?

No media. It was not until the late Qing Dynasty that China had its first medium of communication, the newspaper. Until then, the concept of the "Zhou Ritual" was always followed - the people could be made to know, but not to be made to know. Therefore, the rulers of all Chinese dynasties were not allowed to let the people know the affairs of the country. At most, some rebels organized "folk rumors" in the hope that through the folk word of mouth to start a point of action. If you recognize "folk word of mouth" as a medium of communication, then this is the only one.

By the way, the official means of communication in ancient China was also very backward. From beacons to horses to pigeons, it's been a few thousand years.