There are no Greek Myths relating to the Persians or Persia directly.

I'm not aware of any specific myths involving the Persians but there are several involving the near east: Medea was an easterner, Troy was in modern turkey & hercules' travels took him through the middle east and India.

Also Herodotus' attempt to explain the origin of the Persian wars by drawing it back to the trojan wars. He was trying to show that the east v West conflict had ancient somewhat mythical origin.

I'm not sure your characterisation of the Persians as 'arch-nemesis of the Greeks' is accurate.

The replies so far are inaccurate. In fact, Herodotus (VII. 61) gives an etymology for the name of the Persians:

When Perseus son of Danae and Zeus had come to Cepheus son of Belus and married his daughter Andromeda, a son was born to him whom he called Perses, and he left him there; for Cepheus had no male offspring; it was from this Perses that the Persians took their name.

Perseus is, of course, the hero of Greek mythology who famously killed the gorgon, Medusa, among many other things. Nothing else, however, is known about Perses. Later in Herodotus' narrative (VII. 150), the Persian king Xerxes I sent a herald to the Greek polis of Argos, the mythological birthplace of Perseus. The herald said:

Men of Argos, this is the message to you from King Xerxes. Perses our forefather had, as we believe, Perseus son of Danae for his father, and Andromeda daughter of Cepheus for his mother; if that is so, then we are descended from your nation. In all right and reason we should therefore neither march against the land of our forefathers, nor should you become our enemies by aiding others or do anything but abide by yourselves in peace. If all goes as I desire, I will hold none in higher esteem than you.

So, according to Herodotus, the Persians are descended from Perseus, and the Persian king himself used this myth in an attempt to win over a Greek city during the Persian Wars.

Now, Herodotus is not a mythographer, and his Histories do not contain every Greek myth that existed. But these instances do show at least one myth that circulated in 5th century Greece concerning the Persians. Moreover, the philosopher Plato backs up Herodotus' statements. In Alcibiades I. 120e, Socrates says to Alcibiades:

Then let us consider, by comparing our lot with theirs, whether the Spartan and Persian kings appear to be of inferior birth. Do we not know that the former are descendants of Hercules and the latter of Achaemenes, and that the line of Hercules and the line of Achaemenes go back to Perseus, son of Zeus?

Scholars have presumed that Achaemenes and Perses were likely equated in Plato's mind, since the Persians called themselves the Achaemenid Dynasty, while the Greeks called them the Persians. Hence the two etymologies were conflated. Nonetheless, Perseus, and thus his father, Zeus, were the ancestors of the Persians according to Greek myths.

Somewhat related is the tragedy The Persians by Aeschylus. It is not mythological in its content like the majority of Greek tragedy, but it does add an extra layer to the reception of the Persians in Greek storytelling. It recounts the situation of the Persian court in the capital receiving the news of their defeat at the hands of the Greeks. It is undoubtedly a very enlightening piece of Greco-Persian interactions.