No offense, it's easy to do, but I think you're mistaking driving political events for existing in history. None of these groups disappeared. In fact all of them remained important for at least a little while after being conquered by Cyrus the Great (and all the examples you listed were conquered by Cyrus).

First, a note about terminology. "Mesopotamia" refers specifically to the area around the Tigris and Euphrates Rivers, very roughly modern Iraq. The only place you asked about there is Babylon. Lydia and Ionia were part of westerns Asia Minor or Anatolia, basically modern Turkey. Media's territory extended into only the northern fringes of Mesopotamia, and they were mostly in Northwestern Iran.

I'll start with Lydia because it's probably the least significant of the whole list here.

Lydia was conquered in the mid-540s BCE. No sooner had Cyrus left the territory in the hands of a Persian governor and a local Lydian treasurer, than said treasurer took the treasury and funded the first revolt against the Persian Empire. That was put down and the new Persian general in charge hunted down the instigator before moving on to Ionia (more there in a minute). That's hardly going quietly. After that, the original territory of Lydia was broken up into multiple satrapies. The Satraps in Asia Minor acted as the first Persian point of contact with Greece and usually played a part in Greco-Persian conflict or cooperation. Other than that, they do not really play a major role in politics independent of other events. That's probably becasue Lydia (now the province of Sparda) remained very wealthy and economically important. The capital of Sardis remained a trade hub into Roman times and a close eye was kept on it until the arrival of Alexander. The notable exception to their usual lack of role in imperial politics is the rebellion of Cyrus the Younger against Artaxerxes I (the subject of Xenophon's Anabasks) as Cyrus the Younger was a satrap or viceroy in Asia Minor and the western empire was his power base. As a culture, Lydia continuee to have distinct styles. The tombs of the Lydian royal family were used as models many new nobles in the region as well as (possibly) the that of Cyrus the Great himself. Their language is attested into the 3rd century BCE, after which it was supplanted by Greek.

Ionia is probably the one that least belongs on this list. Technically, "Ionia" is only the Greek cities on the central western coast of Asia Minor, but is often expanded to include the Achaean Greeks north of Ionia and the Greek cities in Caria to their south. In theory they were ruled by Lydia before Persia, but they were largely independent and had to be brutally subdued. That didn't sit well with the Ionians of course, so after a few decades of Persian rule and interference you get the Ionian Revolt. Athenian involvement in that revolt was the instigating event for the Greco-Persian wars (this is grossly simplified). It took a few decades, but the Greek allies in the war against Xerxes actually went on the offensive after the Battle of Salamis and Athens continued that offensive long after Sparta quit the war. In the decades after Xerxes left Greece, Athens and their Delian League "liberated" many Ionian cities and incorporated them into their own de facto empire. A few were were contested by Persia and Athens, but in general they stayed out of Persian hands and played no role in Persian politics because they were no longer Persian. We hear very little about them because we generally don't hear much about any Greek city state other than Athens and its most significant rivals (ie Sparta). If Lydia was the intermediary between the Greek and Persian worlds, then Ionia ended up being the first phase of the Greek world across that border. The most notable casualty of Persian occupation in Ionia is that Miletus was originally the hub of Greek philosophy. The philosophers fled from Persian rule and philosophical thought eventually shifted into Athens. After the conquests of Alexander, these cities became just a few more Greek cities in a vast Hellenistic world and most of them stayed that way into the Byzantine period.

On to Media, the first place conquered by Cyrus the Great. Distinguishing Persia and Media is difficult and may be an entirely superficial project. They were closely related western Iranian cultures and Persia was a Median vassal or subject kingdom. Cyrus' first conquest was really just an overly successful rebellion, helped greatly when Median generals went over to Cyrus' side of the war. The Medes never went away, in fact the Persians are often called "Medes" interchangeably in outside sources. Ethnic Medes were the only group normally allowed into the Persian nobility outside of the Persians themselves, and the only other group that could be appointed as Satraps. One of the royal capital cities, Ecbatana, and several royal fortresses/palaces were located in Media. Generally, Media was a core component of the Persian empire. The main exception was in the time around the ascension of Darius I the Great. In the Behistun Inscription he identifies one Median rebel among the rebellious Satraps in addition to the story of Gauamata the Magos) a Median ethnic/social group) replacing prince Bardiya and usurping Cambyses.

Finally, we turn to Babylon. I think this is the one that you probably have the greatest misconception of. Lydia was important, but not in the geo-political way that makes narrative history. Ionia is important because it tried not to be in the Persian empire. Media was so thoroughly fused with Persia that it was hard for outsiders actually living there to see where one started and then other began (specifically I'm thinking of Ctesias of Cnidus).

Babylon is important because it was Babylon. The fighting over the city itself was short and Persian partisans within Babylon may have helped hand the kingdom over to Cyrus the Great. The primary change for Babylon in the Persian period was that there were new people in charge. Outside of that Babylonia continued as usual. Cuneiform tablets in Akkadian were produced as records, taxes were paid to a distant king, Persian nobles ran the local estates with little disruption, and the satrap fulfilled many of the duties of the old kings. The Babylonian Chronicle continued to record a brief account of what happened each year, dated by the year since the current (Persian) king took the throne. Mesopotamian religion continued to be practiced (with the goddess Ishtar heavily influencing the Persian/Zoroastrian Anahita). The city of Babylon remained a key capital city for the Achaemenids, home to a significant treasury and archive and remained the nexus of several major trade routes. That state continued into Hellenistic times when Greek influence started changing more institutions. When the Seleucids moved the capital to a new city, Seleucia on the Tigris, that really signalled the beginning of the end for Babylon as an important center, but it continued as a city for centuries and traditional Mesopotamian temples continued to operate. One key document being the Babylonica of Berossus, a Babylonian priest duirng the Hellenistic period. Our last Akkadian cuneiform from Babylon is as late as the first century CE. As a political province, Babylonia persisted as just another region centered on a large town or city through Parthian and Sassanid times and was only dissolved after the Islamic conquest. The city of Babylon persisted in Islamic records for another few centuries, gradually shrinking and losing even more prominence as Baghdad rose to prominence. When exactly it completley faded away isn't clear. There are scattered references as late as thr 12th century, but at some point it becomes unclear if it might be an erroneous exonym for Baghdad.

Sources with special attention to Briant, Dusinberre, and Beaulieu.