Evidently, there was no universal time reference that all cultures used, especially given that they had different measurements for years. The most common dating system typically referred to the local rulers, or the emperors, given rise to different calendars.

If we go far back to Ancient Egypt, just to give you a sense of how different dating can be, you can get a sense of how an old civilization like the Old Kingdom kept track of events, like this inscription dated to the reign of Pepi I (c. 2289 - c. 2255) recording an expedition:

Year after the eighteenth occurrence [of the numbering that usually occurred once every two years, but became more frequent during the reign of Pepi I], third month of the third season [Ancient Egypt only three seasons], day 27 of the king of Upper and Lower Egypt [meaning unified Egypt], Merire [throne name of Pepi I], who lives forever; first occurence of the Sed Jubilee [an important royal festival]. Royal commission which the chief of all works of the king, the sole companion, master-builder of the king, attached to the Double House, Merire-meriptah-onekh; his son, the ritual priest, Merire-meriptah-onekh; and the treasurers of the god, Ikhi and Ihu, carried out. [1]

This was the typical dating, using the king's accession to the throne as a reference point. And this continued during the Middle and New Kingdoms (sometimes referring to years instead of the occurrences of the royal jubilee) [2].

The Babylonians had their own calendar, which was lunisolar, i.e. it had 12 months, each begins when the new crescent is observed at sunset, and it was "corrected" to fit the solar year by inserting intercalary months. It is not until the 5th century BC that the Babylonians seem to have codified rules to add these intercalary months: A 19-solar-year cycle corresponded to 235 lunar months (235 = 19*12 + 7), so it was decided to add an intercalary month every first, 3rd, 6th, 9th, 11th, 14th and 17th year of the cycle. In the Inscription of Nebuchadnezzar II, written by the king in a cuneiform cylinder using boastful language, says:

Nebuchadnezzar
King of Babylon,

glorious prince,

[...]

King of Babylon am I.

[...]

A fullness of the treasures of countries I accumulated;
around the city it was placed as an ornament,

when at the festival of Lilmuku at the beginning of the year,

on the eighth day (and) eleventh day,

the divine Prince, Deity of heaven and earth, the Lord god,

they raised within it. [3]

The Babylonian calendar was still being in use by the Achaemenids before it was later replaced by the Zoroastrian calendar (though they don't seem to have followed exactly the rules for intercalation though). The same pattern is also observed in biblical texts, such as the following example:

In the fourth year of King Darius, the word of the LORD came to Zechariah on the fourth day of the ninth month, the month of Kislev. (Zc 7:1)

Things changed a bit after Alexander's conquests. The Macedonian have adopted the Babylonian calendar, but with Macedonian names for months. The adapted Seleucid calendar starts with year 1 which corresponds with April 311 BC to March 310 BC. This calendar was used in much of the Middle East since its inception in the 4th century and continued to be used in Syria (see [4] for a clearer picture).

In Rome, there were a couple of dating systems, like the one referring to the founding of Rome, but this one was rarely used. The common dating system in practice was the consular system, which involves naming the consuls taking office since the 1st of January of a certain year (the consuls for elected for a one-year term). For example, Julius Caesar was assassinated on the Ides of March, which corresponds to the 15th of March, 44 BC. Everyone knew what year it was.

I can't tell you for sure what happened next, but I assume that after the civil war and the establishment of the Empire, a regnal system was in use. For example, I found in [5] transcriptions of Jewish tombstones, like [CIJ ii no.1467], made for a Jewish boy, now found in the Cairo Museum:

Jacob, untimely dead, clever one, farewell. About 11 years old. In the 3rd year, Thoth 14.

The part "In the 3rd year", refers to the accession of a ruler, here thought to be Augustus (so, around 24 BC). Thoth is an Egyptian month.

The term anno Domini (AD) wouldn't be used for the Julian calendar year until later in 525 AD.

----------------

References:

[1] W.R. Harper, Ancient Records of Egypt. Vol. 1, p. 137 (§ 298).

[2] See for example Ahmose I's Quarry inscription in W.R. Harper, Ancient Records of Egypt. Vol. 2, p. 12 (§ 27), referring to "Year 22 under the majesty of the king, son of Re, Ahmose". Or in p. 22 (§ 51), referring to the 10th year of Amenhotep I.

[3] Inscription of Nebuchadnezzar, in Babylonian and Assyrian Literature, Translator: Rev. J. M. Rodwell, M.A. P. F. Collier & Son, New York, 1901. See text here.

[4] T. Boiy, Local and Imperial Dates at the Beginning of the Hellenistic Period, first chapter in the book E. Dabrowa, New Studies on the Seleucids. Also, B.L. McLean, An Introduction to Greek Epigraphy of the Hellenistic and Roman Periods from Alexander the Great down to the Reign of Constantine (323 BC - AD 337).

[5] W. Horbury & D. Noy, Jewish Inscriptions of Graeco-Roman Egypt. p. 124.

You might enjoy this thread from the FAQ, which answers your question for Classical Greece!