How did people measure, mark, or denote years before the year "0" (B.C., A.D.), as they could not have predicted the "birth of Christ" to denote B.C. years?

Before the AD/BC calendar came along, there were many methods used by cultures to keep track of years.

The most common way was to record things as happening "In the eighth year of the reign of King/Chief Mogwai": the regnal year system.
The Republican Romans used a variant of the regnal year system, based on consular years ("the year of Consuls C. Iulius Caesar and M. Antonius"). Before the Republic, they used the regnal year based on their kings. And, by the time of the Emperor Justinian, the Romans were counting years "in the reign of Emperor...".
Another alternative was to start counting from the year the world was created, like the Jewish calendar.
Yet another way of counting the years was in cycles, like the Mayans.

You may be interested in the "Other dating systems" section of the Popular Questions pages (as found in the sidebar).