“May the day bring me health, the month joy, the year its abundance”

Sumer was the first civilisation to create a calendar, which was later adopted by the Hebrews, Egyptians, Hittites and Akkadian civilisations.
The Sumerians equated the calendar by 12 lunar cycles of the annual agricultural cycle. As they used the phases of the moon, they calculated 29-30 days per lunar month, thus: 30 * 12 = 360
As with even the modern day calendars, the months were not completely accurate, and approximately every four years (or so) an extra month would be inserted into the calendar
The days of the month would begin and end at sunset, containing 12 hours (ie. 6 hours of daylight, 6 hours of night)
As this was the earliest civilisation, there were no true names assigned to the lunar months until the occupation of Sumer by the Babylonians in the years to come

The Babylonian names assigned to the months are as follows:

Nisanu “First”
Ayaru “Rosette/Donkey Festival”
Simanu “Season”
Du-uzu “Dumuzi/Tammuz”
Abu “Mound over the passageway to the Netherworld”
Ululu “Purification”
Tashritu “Beginning”
Arakhsamna “Eighth Month”
Kislimu
Tebetu
Shabatu “Blowing Storms”
Adaru