Akitu is the ancient Mesopotamian festival often linked to the historical roots of modern New Year's celebrations.
Akitu was a major Babylonian and Assyrian festival connected with the renewal of kingship, the agricultural cycle, and the spring new year. In Babylon, it was celebrated during the month of Nisanu, corresponding broadly to March or April rather than January. The festival included processions, religious rites, and ceremonies involving the king and the god Marduk.
The Babylonian New Year was not a direct, unbroken version of today's January 1 celebration. Modern New Year's customs developed through many later calendars, religious traditions, civic practices, and local festivals. Akitu is important because it shows that public year-renewal ceremonies existed in the ancient Near East and could combine ritual, politics, agriculture, and community identity.
Saturnalia was a separate Roman festival associated with Saturn and usually held in December. Panathenaia and Dionysia were ancient Greek festivals with different religious purposes, so they are not the best answer to this Mesopotamian New Year question.