Recently I read a blog advocating that Numbers 24 may contain material dating to the Greek or Roman periods of Judean history. Does it?
The argument that it does is based primarily on interpreting Kittim to mean Greece or Italy as it does in 1 Maccabees or the Septuagint Daniel. If this is so, then what is the significance of “Kittim” to identify them? The word comes from Kition,  a town in Cyprus well known in antiquity. It is used in this sense in Isaiah 23:1. While the name of a town, it was sometimes used as a generic term for sea-faring nations of the west. The view that it is late interprets the Oracle of Balaam in Numbers 24:15-24 to be a prophecy of the Greek colonization of the Levant after Alexander the Great’s time. The Greeks afflict Asshur and Eber, that is Assyria and Israel, and in turn are destroyed (by the Maccabees?).
The person this prophecy is attributed, Balaam son of Beor, is however independently attested to in the region of modern Jordan with material dated to the 8th century BCE. Other than the accounts in the Bible, this character is unheard of in the record. Since he is unattested to in later culture, it suggests an earlier period for the Balaam material in the Bible, including this poem. It is not impossible that Biblical texts were altered during the time of the Maccabees, and it is in fact likely they have been. Is Numbers 24:15-24 among them, created as evidence that the victory over the Greeks was prophesied, like Daniel? It seems more likely that instead the poem is a prophecy relating to the early history of Israel, before Solomon, composed at an early stage of the Israelite history tradition. That is earlier than the account in Samuel and Kings. This is supported by the language of the text, which doesn’t betray the influence of Greek or Persian styles and terms.
If Kittim means Cyprus and people from the western seas generally, were there any peoples fitting this description earlier than Alexander’s Greeks? Yes there is, the Philistines were part of lager migration of people some coming from the western Mediterranean, Greece and the Aegean islands. They were active in the region from the 12th century on.
Eber, seems to clearly refer to the Eber who is the ancestor of the Hebrews, and thus the Hebrews themselves and is sometimes a synonym for the more popular term Israelite, though properly, Hebrew would include a number of southern Semitic groups. Asshur could either refer to Asshur the mythic ancestor of the Assyrians as in Genesis 10:8-12  or the Asshurim from Genesis 25:1-3 who’s is among the Arabian sons of Abraham. The Kenites and Amelkites mentioned in verses 21 and 20 are two groups associated with each other in 1 Samuel 15:6, where Saul eradicates the Amelkites and drives the Kenites from their homes. Edom and Moab are both well-known biblical nations. The prophecy of the subjugation of these groups reflects well the emergence of the Davidic state as described in the Primary History (Genesis-2 Kings).
Rather than being a cryptic metaphorical description of the state of affairs under the Greek kings, it is more naturally a description of the events described in the Primary History, but from an earlier age when groups like Amalek, Kain, and Asshur (the Arabian Asshur) where more prominent. This scenario does not require the metaphorical meaning of the names as required by the theory placing it later under Greek influence. It also places it closer in composition to the other known example of a prophecy related to Balaam son of Beor. Finally, the Oracles of Balaam contain no other indicators such as Persian or Greek Language, or clear references to historical facts of the Greek or Persian period that would lead one to suspect it was composed during the time of those empires control of the land of Israel.
 “It is significant that the first biblical definite evidence of the existence of the biblical texts is in the Hellenistic period — after the conquests of Alexander the Great and the spread of Hellenization (Greek culture and language) throughout the Middle East.
Lemche stresses that a justifiable dating of a text must come from the latest material found within it, even if it does contain some earlier material. So if the Pentateuch contains passages that can only be explained as of Hellenistic provenance, then the Pentateuch as we have it surely must date from Hellenistic times even if it contains some Mesopotamian material (e.g. the flood story) that is earlier. Lemche points out, that Karl Ilgen first told us 200 years ago that Numbers 24:24 can only possibly be a reference to the Macedonians and that Martin Luther had likewise made this clear in notes to his translation — hence the poem must be dated to the late fourth or third centuries BCE. Yet despite this clear pointer to a late date scholars have generally dated this chapter as some of the most archaic of poetic literature, and certainly pre-monarchic. The chapter was later moved to the monarchic period when the name Balaam turned up in an 8th or 7th century Aramaic inscription. Meanwhile, the poem itself keeps murmuring, “Hellenistic”. (p. 160, Israelites in History and Tradition).” From, http://vridar.wordpress.com/2012/01/08/who-wrote-the-bible-2-challenging-the-documentary-hypothesis/