Bricker asked specifically about the ancient Middle East, apparently under the misapprehension that most modern languages spread from there.
The problem lies in the fact that when one discusses language, one is looking at a large number of language phylums. Conventional wisdom is that there is no clear evidence of any interrelationship between those phylums, though hypotheses identifying their relationship have been advanced.
Modern Europe is dominated by Indo-European languages, the sole exceptions being Basque, Hungarian, Finnish, Turkish, Estonian, and a number of minority-group languages spoken in European Russia. These fall into several families such as Germanic, Romance, Slavic, etc. Also in this group are the Iranic group including modern Iranian and Tadzhik, and the Indo-Aryan group spoken in much of India and Pakistan along with the neighboring countries.
Another completely separate phylum is one traditionally called “Hamito-Semitic” but now more commonly known as “Afro-Asiatic,” of which Semitic is the key family for purposes of this discussion.
In addition, tongues like Basque, Japanese, and the Siberian tongue Kamchadal are considered language isolates, with no identifiable relatives.
Now, to look at the Middle East about 1900 B.C.E.
First, the majority culture in southern Iraq spoke Sumerian, a language isolate now extinct and with no known relatives. The area to the east, in coastal Iran, spoke Elamite, another language isolate. North of the Sumerian area in Iraq was spoken Akkadian, of which Babylonian and Assyrian were dialects. This was a Semitic language. Another group of Semitic languages included Ugaritic, Phoenician, Hebrew, Aramaic, Amorite, Moabite, and related tongues. These were spoken in the Western Fertile Crescent, roughly Israel, Jordan, Lebanon, and Syria Of these, only Hebrew and Aramaic survive, with Hebrew artificially preserved by Israel and Aramaic on the verge of extinction. Old dialects of Arabic existed at this time but were confined to the central part of the Arabian peninsula. And “South Arabian” languages were widespread across present Yemen and neighboring areas in both Arabia and East Africa, with relatively small surviving groups still speaking them in Arabia and Ethiopia.
Ancient Egyptians spoke, not surprisingly, Ancient Egyptian, part of a distinct family within Afro-Asiatic. It survived until relatively modern times as a spoken language, though replaced by Arabic for the most part in the later first millennium C.E.
Ancient Greek and its sister language Phrygian occupied the Greek peninsula and islands and parts of the Aegean coast of Anatolia (modern Turkey).
The majority of what is now Turkey, however, spoke languages belonging to the now-extinct Anatolian family of Indo-European, including Hittite, Lycian, Lydian, and related tongues. Two other languages, Hurrian and Urartuan, were spoken east of the Anatolian group, and may have been sister languages in that group or isolates; the evidence is very unclear. They were replaced eventually by Armenian, a separate family in the Indo-European group, which was actually significantly more widespread than its present usage area before the 20th Century.
Elamite was fairly early replaced by Old Persian, which coalesced with Avestan, spoken north and east of the Persian homeland in the southwestern Iranian mountains, to produce modern Iranian and related tongues.
The modern state of affairs, in which most Iranians speak Iranian, most Turks speak Turkish, and most other Middle Easterners speak Arabic, dates from the spread of Islam, carrying Arabic with it, in 600-900 C.E. and the invasions of the Seljuk and other Turkish peoples in 1000-1500 C.E.
This omits a lot of detail, but it gives a good rough summary of what is generally known about Middle Eastern language history. In general, it can be said that almost nowhere in the Middle East is there an area of land where people today speak a descendant of what was spoken in that place in Abraham’s day, with continuity of that language in that place straight through nearly 4000 years of history.