Where does the tradition come from that Noah's sons went separate directions? I have heard in many places, and was even taught as a child, that Noah's sons, sometime after the Flood, separated and went North, South, and East to repopulate the Earth. The story is commonly used to point out that North, South, and East of where the Ark is said to have settled have three distinct races, which are those of Europe, Africa, and Asia, as this answer mentions and AiG hints on.

The issue is that I cannot find this in the Bible. I see the nations that come from Noah's sons listed in Genesis 10, but that doesn't say the sons went in any particular direction. The story of the Tower of Babel follows in Genesis 11, after it is implied that it has been some time since Noah's sons had died.

From a Google search I found this very detailed description, but there is no source.

Since this doesn't appear to be Biblical, where did this story, or tradition, or theory, come from.
See Josephus' Antiquities Book 1, Chapter 6. He gives a rather detailed interpretation of Genesis 10, identifying what nations he believed the Hebrew terms refer to.

Japhet, the son of Noah, had seven sons: they inhabited so, that, beginning at the mountains Taurus and Amanus, they proceeded along Asia, as far as the river Tansis, and along Europe to Cadiz; and settling themselves on the lands which they light upon, which none had inhabited before, they called the nations by their own names. For Gomer founded those whom the Greeks now call Galatians, [Galls,] but were then called Gomerites. Magog founded those that from him were named Magogites, but who are by the Greeks called Scythians. ....
The children of Ham possessed the land from Syria and Amanus, and the mountains of Libanus; seizing upon all that was on its sea-coasts, and as far as the ocean, and keeping it as their own. ....
Shem, the third son of Noah, had five sons, who inhabited the land that began at Euphrates, and reached to the Indian Ocean. For Elam left behind him the Elamites, the ancestors of the Persians. ....

The quotations (and the text at the link) are the Whiston translation, which is unfortunately written in rather archaic syntax.