I am assuming that by asking whether or not they were built you mean to ask whether or not they were ever in operation? The actual cities themselves were certainly built. They comprise three cities West of the Jordan (Kedesh, Shechem and Kiriath-arba - Joshua 20:7) and three cities East of the Jordan (Bezer, Ramoth-Gilead and Golan - Deuteronomy 4:43 and Joshua 20:8). Later on, these same cities are listed as Levitical cities (Joshua 21 and 1 Chronicles 6), as foreshadowed in Numbers 35:6.
Some scholars have questioned the historicity of these cities as Levitical cities, but I don't think anybody has ever questioned the operation of the cities of refuge. Since the six of them didn't always all fall within the borders of the land of Israel, dates for their construction and operation vary. If you're interested, John Spencer has an article on this in The Anchor Bible Dictionary, under "Refuge, Cities of" (vol. 5, pp657-658), with a bibliography. That's not a "traditional" source, obviously, and how useful it will be for you depends on your general approach.