You touched a sore spot, because there really isn't anything in Chinese that's comparable with Homer's work.
The reason, I think, is that in ancient China, history logging is a monopoly business. It's always in the hands of the royal families and their ministers. There might be great story tellers among the common folks, but the literate class just didn't bother to record them down. Or simply, maybe we just didn't have the luck, so no such great artist was born.
But obviously we did have an active oral tradition. Besides 诗经, 木兰辞 is one, 孔雀东南飞 is another, but there is nothing epic about them.
《山海经》 for example, recorded myths that trace back to prehistory. These are, however, just short stories.
《聊斋》 is literally folk stories told by men on the street, but again, they are just short stories, and nothing historically important.
《格萨尔王传》 is something.. oops, it doesn't belong to the Sinosphere. Similar are some some other epics produced by the minority cultural groups. Anyway, we conquered and ruled them, so they belong to us now. I've even read Korean folk tales when I was young.
Regarding to the oral tradition of the Han ethnicity, we have 评书, 快板, 大鼓, etc. These things are probably very ancient. Many of them are still pretty much alive. They all tell great poetic stories about the past. 三国演义 and 水浒 are probably of such origins, they are pretty epic to my standards, but not ancient enough to compare with Homer.
I have a deceased elder relative who was in the business of 说书. I used to follow him when I was little. Like Homer he was blind. His instrument was 三弦 and 惊堂木. When he visited a town or village, people would gather at small theaters or open places to hear him talk and sing about the past. Oh the good old days....
中国为什么没有史诗 by 朱光潜
Also from this interesting discussion:
Gilgamesh: I hunted monsters.
Rama: I killed the demon king to save my wife.
Gesar: Me too.
Achilles: I battled Troy and killed their general with a spear.
Beowulf: I killed dragons.
Siegfried: Me too!
后稷: I cultivated rice and beans....
Ignore those who say "Because Chinese culture is superior!". They are stupid.