The Kurdish nationalist attempts to construct the myth of "ancient origin" resemble very much those of nationalist movements in the Caucasus, Central Asia and the Balkans, especially their autochthonous narratives. In all such cases, alleged ancient origins are based on pseudo-etymologies, outdated theses, anachronisms, historical falsehoods and cultural usurpations. The myth serves three basic purposes: first, it creates the impression of a long and glorious past, second, it implies that specific ethnic group had cultural self-awareness and inhabits a particular territory since time immemorial, and third, it serves as a base for further claims of an unbroken historical continuity between the ancient and the modern people. On this point, we're not speaking of national movements which are in most cases the nineteenth-century inventions, but of the concept of ethnicity as a group of distinctive cultural traits and known by the specific name.
So, who are the Kurds, where do they come from, and how old they are? As an ethno-linguistic and cultural group, they are undoubtedly classified as one of the Iranian peoples (or Iranic), along with the Persians, Pashtuns, Baloch, Lurs, Mazanderanis, Gilaks, Ossetians and other smaller groups. Proto-Iranians have emerged as a separate group in the mid 2nd millennium BC, and in next centuries they expanded from Central Asia across the Iranian Plateau (south) and the vast Eurasian Steppe (north). The high level of cultural similarity between those who went toward south is owed to the long-standing, unified and centralized Iranian empires since the beginning of the 7th century BC. Their territories continuously included the central Western Asia area, known as Kurdistan today.
Beside the ancient Iranian peoples, among who only the Persians existed under the present name, in the 1st millennium BC area was also inhabited by Armenians in the north, the Semitic peoples (including the Arabs and Assyrians) in the south, and several mountain tribes whose origin is not known. Speaking of the existing ethnic groups, the Turkic tribes arrived there latest, around the 11th century. Of course, it would be erroneous to assume that the characteristics of those ethnic groups were fixed and enduring, since they all interacted, mixed and assimilated parts of others, but there is no doubt the Arabs, Armenians, Assyrians and Iranians (including the Persian subgroup) are attested in ancient period under contemporary names. However, that's not the case with the Kurds.
The term "Kurd" first appeared in Muslim travelogues and other books from the early Middle Ages, and is conceivable the root of the word is of Persian origin. This label is given by non-Kurds, mostly Arab and Persian authors, as tribes of that time ignored the name "Kurd" and called themselves by their tribal or clan name of the particular micro-region or valley they were living in, or from the mountain along which they were nomadizing. The usage of the term by the Medieval authors was of purely social character, rather than a concrete ethnic or linguistic group living on fixed and homogeneous territory. Their accounts describe Iranian nomadic and rural tribes who inhabit the same places as the Persians and Arabs, and stated that all these people spoke "Persian" language. This all confused Medieval geographers and writers who claimed they may be either of Persian origin (based on language), or Arabic origin (based on a way of life).
Interestingly, a roughly defined "ancestral homeland" of those proto-Kurds wouldn't be a present-day cultural region called Kurdistan, but mainly the inner Iranian plateau, and this explains why contemporary Kurdish languages share the highest similarity to the Balochi language, spoken in the region some thousand kilometers southwest. It is noteworthy that the early Medieval authors did not use the term "Kurdistan" to denote a specific territory populated by the Kurds, they instead applied geographical appellations or expressions like Jibal (Mountains) or Zozan (Summer Pastures). The word "Kurdistan" itself was first employed by the Turko-Persian Seljuk Empire in the 12th century, as a term for an administrative unit located in the eastern part of the Zagros Mountains near Hamadan. Again, this region in the earliest references has been purely of an administrative nature and did not concern itself with the ethnic, linguistic or cultural boundaries, for it left out some other major towns populated by the Kurdish tribes.
The fact that the Kurds found themselves without an ancient history has been very frustrating for their nationalists who, inspired by European-style primordialism, desperately seek to construct ancient roots and create a national culture. Their romantic nationalism parallel that of the Turkish and Balkan nationalists, because all of them adore the outdated theses based on pseudo-etymologies. No matter which kind of illusory correlation is used, it can be an unidentified archeological culture found on modern ethnic-dominated territory, a pseudo-ethnonym written on a Sumerian inscriptions or even an Indian mythological river a few thousand kilometers away, the joy of deepening own history overcomes all scientific obstacles.
In the Kurdish case, it is often claimed that their ancient ancestors are "Qarda" found on a Sumerian clay tablet from the 3rd millennium BC, or "Karduchi" and "Cyrtii", Mesopotamian mountain dwellers mentioned by several classical authors. These proposals were given by foreign (mostly Western) scholars over a century ago, based on some philological similarities with the word "Kurd", but all have been dismissed for decades because these words have foreign roots, not to mention lack of continuity in sources and chronological holes of few thousand years between them.
Another popular myth suggests that the Medes, an ancient Iranian people who inhabited the land where currently the Kurds are the majority, are their forefathers, and such claim has even been incorporated in the national anthem of Iraqi Kurdistan. Still, this outdated view is rejected by leading Kurdologists due to linguistic differences and a historical fact of migrating tribes described in the Middle Ages as the Kurds. There is no doubt that ancient Iranian tribes influenced medieval ones by both genesis and culture, however, the Kurds are not the only medieval or modern Iranian people living on ancient Median land. In fact, half of ancient Media was later inhabited by Azari people whose medieval Iranic language (before Turkification) was closer to Median language.
There's also no doubt that earlier prehistoric tribes influenced proto-Iranian peoples, including ancestors of the Kurds. Those indigenous peoples did not disappear because they became extinct or exterminated, but because they were integrated into new ethnicities. However, taking them as a part of "Kurdish history" inevitably implies the acceptance of other autochthonous myths, like the Hittites as a part of "Turkish history" or the Sumerians of "Iraqi history." Their nationalist pseudo-histories share identical framework and involve even similar pseudo-etymological claims from clay tablets (Turukku and Uruk). In all three cases, beside anachronistic territorial relation, there's absolutely nothing cultural in common.
Summarized, Kurdish nationalist discourse regarding "ancient origin" is not only historically incorrect, but can be highly counterproductive when used for justification of territorial pretensions and monopoly based on alleged "historical right." It has been constructed in purpose of proving the Kurds are the oldest inhabitants in "Kurdistan," however, as a roughly defined ethnic group they appeared in these areas only before the Turks. Of course, this does not justify any other nationalist argument directed against either Kurds or Turks, especially if it's exclusive in nature like Zionist anti-Palestinian ones. One-sided political decisions have always been dangerous, particularly if based on selective historical, political and demographic arguments.