The Achaemenids may have been long gone, but their cultural, religious and social descendants, the Sasanians, were still around until the early 700's, though reduced to a rump state of sorts in Sistan after the Arab conquest of their territory in the mid-7th century (it wasn't exactly a rump state - the Sasanian royal family fled to China and reconquered some territory with an army they were granted by Emperor Gaozong). One of their subject territories, the Karenid kingdom of Khorasan, survived with relative independence until the mid-800's.

The degree of continuity between the Achaemenids and Sasanians, besides the obvious fact that they were Persian Zoroastrians, is still disputed, however. Notwithstanding numerous literary, epigraphical and ideological parallels, and the fact that the Sasanians chose to insert their inscriptions and reliefs near those of the Achaemenids (which were impressive regardless of the degree of memory retained) the only direct evidence of a known connection is the use of a term, "ahenag" or "hasenag", by the second Great King Shapur, on his inscription at Kaba-ye Zardusht. This term, describing the Sasnians' alleged forerunners, have been identified either with the Achaemenids or their epic counterparts, the Kayanids (known from the Shah Nameh, based on the lost Middle Persian compilation the Khwaday-Namag). A more recent trend, identified by MR Shayegan in Arsacids and Sasanians, is to actually identify the Kayenids with the Achaemenids, as the Sasanians remembered them.

Persian and generally Iranian oral epic has a notoriously short memory when it comes to details, and frequently jumbles names, events, and individuals. It is possible, for example that one or both of the two longest-reigning Achaemenid kings Artaxerxes I/II (whose birthnames were Arshaka/Arsaces) had been conflated with Arsaces, the founder of the "Parthian" (Arsacid) Empire; one classical source, which by virtue of being written down had a far better memory of Persian history than the Persians themselves, notes that the "Parthian" Arsaces took this name because he traced his descent to an unspecified Artaxerxes [I will note here: Parthian was the official language of the Arsacid Empire, but Arsaces was not a Parthian, hence the variations in terminology]. In any case, Arsaces was remarkably popular to trace one's descent to (many noble families in the Sasanian Empire did so; the Sasanians did not, since they defined themselves by their revolt against the Arsacids), especially when considering how modest his conquests were (the great conqueror of the Parthians was Mithridates I, who reigned fifty years after the death of Arsaces).

The degree to which the Arsacids inherited Achaemenid tradition has also been disputed; Shayegan in the aforementioned work (which I have yet to read more than a few dozen pages of) claims that his investigations required a massive revision of our view of Parthian political culture. Generally it has been noted that the Arsacids were stylistically very Hellenic, by virtue of being the immediate successors and conquerors of the Selecuid Empire established by one of Alexander's generals. Unfortunately, they left remarkably little to investigate them through, despite flourishing for nearly five hundred years; they certainly seem to have been less interested in the use of reliefs, monuments and inscriptions for propagandistic purposes than the Sasanians and Achaemenids were. If I recall correctly, more inscribed text survives from either from the Achaemenid Darius I or the Sasanian Ardashir I, than from the entirety of the Arsacids' existence. Still, they practiced a variety of Zoroastrianism and spoke a language very similar to Persian, so there was presumably at least some cultural memory from the Achaemenid era, which had ended not a century before the conquests of Arsaces.

It is interesting that you mention Aramaic. Imperial Aramaic was the lingua franca of the Achaemenid Empire (I am not sure if it ever acquired the ceremonial/official significance of Old Persian, Elamite or Assyrian, though) and Arabic actually inherits a number of loanwords for it (e.g. majus from Old Persian magush; the stem -ush disappeared in Middle Persian since nouns were no longer inflected), and several North Arabian kingdoms were under Achaemenid suzerainty from the conquests of Cambyses. While the beautiful cuneiform script of Old Persian was lost, perhaps because it was designed for stone inscriptions and impractical for clay and paper, the Aramaic script lived on through the Parthians and Sasanians; the Middle Persian Pahlavi script (corruption of Parthavi, essentially meaning "Parthian script") is a modified form of Aramaic. Indeed a lot of ideograms in Aramaic were still used by the Parthians and Sasanians - for example, "Shahan Shah" is written with an ideogram for "Melekin Melek", which is of course Aramaic for "King of Kings"!

And Sasanian tradition wass certainly not lost to the Muslim polities east of the Tigris. The Buyids, and countless dynasties after them, adopted the Sasanian royal styling "Shahanshah" (King of Kings), and others picked up the related "Padishah" (can be translated "Lord King", but derives from a Middle Persian word pādixšāyīh, "sovereign", often used in the religious-ideological term hu-pādixšāyīh, "good rulership"). The degree to which Zoroastrian tradition survives in Shi'a Islam (e.g. a focus on messianism and semi-divine kingship) is disputed, but indeed the very use of "Iran" (short for Iranshahr, "Dominion of the Aryans" in Middle Persian) for a polity (first officially used by Ghazan Khan, "Padishah of Iran and Islam", as far as I know) is directly descended from Sasanian ideology, though one should be careful to not draw a direct equivalence as many Iranian nationalists are wont to do. Persian oral tradition lived on in works like "A Thousand and One Nights" (indeed the framing story concerns "Shahryar", a "Sasanian King"... in "China and India", but still), and of course Zoroastrian converts to Islam played many important roles in the administration of the early Caliphates.

One could go on for a very long time about the degree to which there was immediate continuity between Persian and Arabic-Islamic tradition, and Arabic writings on Persians and Zoroastrianism, of course, and I suspect that's not primarily what you are interested in (and I'm not very well versed in early Arabic literature anyway). However, I hope this writeup helps to show that there was considerable continuity of ancient empires (not only the Achaemenids, but the many other empires - Assyrian, Urartu, etc, that they were heavily influenced by) right up to the Arab conquests themselves.

Sources and further reading:

Encyclopaedia Iranica (first-stop for all things Iranian history)

Avesta.org (has many Middle Persian writings)

Arsacids and Sasnians by MR Shayegan (2011)

Wiley-Blackwell Companion to Zoroastrianism (2015), chapter on Zoroastrian influence on Islamic writings.

The Persian Empire: A Corpus of Sources from the Achaemenid Period (2007) by Amélie Kuhrt

A State of Mixture: Christians, Zoroastrians and Iranian Political Ideology in Late Antiquity (2015) by RE Payne

In God's Path: The Early Arabic Conquests and the Creation of an Islamic Empire by Hoyland