For within Mesoamerica, the start of the Maya Long Count has been calculated to be August 11th, 3114 B.C. However, the earliest confirmed Long Count Date dates to 31 BC at the Olmec site of Tres Zapotes. Dr. Ed Barnhart runs a neat little website that explains the Long Count in more detail.

I'd like to recommend our podcast episode on the subject.

While there are several ways of counting years in Jewish history. The most common one today, years from the biblical creation narrative, while unrelated to Christianity, was actually developed after Christianity existed. So it's non-Christian, but not technically pre-Christian (even if its development has nothing to do with Christianity).

The dominant dating system before was years from the Seleucid Era. People would count from the establishment of the Seleucid Empire, following Alexander's conquest. This was common throughout the Middle East, and persisted through the Middle Ages.