While it could be argued that Epicurus and certain other Greeks, like Democritus, have contributed to the foundations of modern secularism, I'm pretty sure that they can't be regarded as the sole source of such.
One shouldn't disregard the eastern schools that have a long history of Atheism and Agnosticism.
While I don't know much about their ideas, the Chinese delved into these matters as far back as 7th Century BCE
Religious beliefs in Ancient China
Concerning Indian schools of thought: There were many such schools, whether they stick to the Vedas or not, that were clearly quite atheistic or agnostic.
One group that stuck out, though, were the Charvakas. They were strict materialists, to the point of Hedonism. Their philosophy is thought to have originated around the 6th Century BCE. While the Charvaka philosophy seemed to have died out around 1400 CE, many of the ideas, principles, and refutations of spiritual thought still exist today in modern atheism. In fact, their main goal was the pursuit of happiness (and pleasure) as a means of reducing suffering.
The Charvakas
"The good Christian should beware the mathematician and all those who make empty prophecies. The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and to confine man in the bonds of hell." - St. Augustine
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