Jeffcoat provides a strong argument that such Greek musical terms would be available to Daniel:
There are numerous considerations that assure the possibility of the use of Greek musical instruments at Babylon in the sixth century B.C., including the following: Greek inscriptions of Abu Simbal in Upper Egypt (dating from the time of Palmtek II in the early part of the sixth century B.C.); the Minoan inscriptions and ruins in Crete; the revelations of the wide commercial relations of the Phoenicians in the early part of the first millennium B.C.; the lately published inscriptions of Sennacherib regarding his campaigns in Cilicia against the Greek seafarers (telling about having carried Greek captives to Nineveh about 700 B.C.) to which Alexander Polyhistor and Abydenus both referred; and, the confirmation of the wealth and expensive ceremonies of Nebuchadnezzar (made possible by the discovery of his own building habits and other inscriptions) [see Wilson, 1939, p. 785].
The term kitharis was used in the epic poetry of Homer (Iliad III, 54; XIII, 731; Odyssey I, 153; VIII, 248) long before Daniel's time (see Tisdall, 1921, p. 208). It does not seem unlikely that, if one [original emphasis] Greek musical instrument had become known in Babylonia before Daniel's time, two others should have been introduced also, especially as the names of other instruments mentioned in the same connection were known not long afterwards in Greece.
As early as the reign of Sargon (722-705 B.C.) there were, according to the Assyrian records, Greek captives who were sold into slavery from Cyprus, Ionia, Lydia, and Cilicia. The Greek poet Alcaeus of Lesbos (600 B.C.) mentioned his brother Antimenidas serving in the Babylonian army. Thus, it is evident that Greek musical instruments were in use in the Semitic Near East long before the time of Daniel (see Archer, 1964, p. 387). The name of such instruments usually does not change from nation to nation. So, it is not unusual that the instruments that are mentioned in Daniel 3:5 should keep their Greek name (see McGarvey, 1956, pp. 259-260).
Furthermore, if the Jews were required to furnish music (see Psalm 137:3), it would not be incredible to assume that Greeks from Cyprus, Ionia, Lydia, and Cilicia were required to do the same (see Unger, 1951, p. 399). R.K. Harrison has observed that the instruments under consideration were of Mesopotamian origin (1969, p. 1126).
Jeffcoat, 'The Linguistic Argument For The Date Of Daniel', pages 5-6, 2004
What is more remarkable than the presence of three Greek words in Daniel is the fact that there are not more of them. According to common criticism, the book was not written until the Maccabbean era, under the Greek rule. Jewish works produced at this time betray their date and authorship by the conscious and unconscious use of the language contemporary with their author.
The book of Daniel does not share this feature - none of it is written in Greek, and the three Greek words which are used a transliterated as loan words.