It seems to me that the Achaemenid dynasty has not been as celebrated as later dynasties in Iran. For example, the Achaemenids barely appear in the Shahnameh while there is a long section on the Sassanids. In a link posted a few days ago, some Iranian playing cards from the 1930s display various Persian dynasties, but not the Achaemenids.
Compare this to the West, where (at least in my experience) I only learned about Achaemenid Persia in school. Any historical films with Persians will always have them be Achaemenids (think 300, even as fanciful as it was). As another example, in the Civilization games, the Persian civilization is only Achaemenid Persia and nothing in the Persian civilization alludes to any other period in Persian history.
I would hazard a guess that it has a lot to do with religion. The Safavid dynasty made Shia Islam the state religion in large part to maintain a distinct and independent identity for Persia, which had historically been repeatedly subjugated by Sunni Muslim neighbours since the 7th century. Since the national identity was built around Shi'ism, there was likely little sense in emphasising any pre-Muslim origin of the Persian "nation".
Mohammad Reza Shah Pahlavi, the last Persian monarch, shifted gear and embraced the Achaemenid heritage whole-heartedly. He took on the title "Light of the Aryans" and even the ancient Achaemenid imperial title of Shahanshah, King of Kings. However, his reign was cut short by the Islamic Revolution, of course.
It's because of the Achaemenid association with Alexander and the Greco-Persian wars. In order to teach Alexander and ancient Greece, you have to teach about the Persians, who are the "enemy", over whom Alexander (the representative of "the West) prevailed. Thus the image of the Achaemenids stays in the Western psyche as one of the first "enemies" which the West conquered.
In short, it's a result of a euro-centric brand of teaching in the West.
It's because of the Achaemenid association with Alexander and the Greco-Persian wars. In order to teach Alexander and ancient Greece, you have to teach about the Persians, who are the "enemy", over whom Alexander (the representative of "the West) prevailed. In addition you see other events like the Battle of Salamis touted as "the battle that saved Western civilization". Thus the image of the Achaemenids stays in the Western psyche as one of the first "enemies" which the West conquered and repelled.
However when it comes to the later Perso-Roman Wars, which lasted 800 years and ended up with no clear victor,
In short, it's a result of a euro-centric brand of teaching in the West. I'm not trying to put a negative implication on that either; it's just the way that Greece is traditionally viewed and taught in the West.
The answer lies in the fact that you've only read Ferdowsi's Shahnameh. Firstly, he does refer to them, as Kianian, which is supposedly the correct term. He refers to the Median Empire as Pishinian. But Abureihan Biruni in Asar Al-Baghieh refers to them all, so does Abu Mansuri's prose Shahnameh.
You see the Macedonians destroyed Persopolis, back then most Iranic people were nomadic, they had little culture compared to the Greeks.