The first "dice" were actually worked knuckle-bones of sheep/goats and cows. They were worked to make them flat on a couple sides and were called kisallu in Akkadian, in modern terms we call them "astragali". The bones were marked with values, and games were played depending on which values came up. The knuckle bones were used because they already nearly had a rectangular shape and little effort was required to make them into the required gambling pieces.

Eventually this was refined into the recognizable cube marked with dots that we know today. Interestingly, this occurred first in modern-day Pakistan, in the Indus Valley civilization and the city of Mohenjo-Daro. The precise date of this civilization is controversial, but is between 2500 BC and 1906 BC, making the modern form of dice about 4000 years old, at minimum. Interestingly, the numbers on the earliest dice were such that opposite faces were numbered sequentially, i.e. 1-2, 3-4, 5-6, while modern dice are numbered such that opposite sides add up to 7, i.e. 1-6, 2-5, 3-4 (seeHallo, William W., and ויליאם ו. האלו. "מקומה של הספרות בהוראה באבלה ובמארי/GAMES IN THE BIBLICAL WORLD." ארץ-ישראל: מחקרים בידיעת הארץ ועתיקותיה (1993): 83-88.‎ (don't worry, it's in English and on JSTOR).

The next oldest are from Mesopotamia. As dice were used throughout ancient history, they were not exactly standardized. The numbers changed a little bit, one of the popular patterns in both Mespotamia and the Indian subcontinent was 1-2, 3-5, 4-6. In some regions, dice were not cubic, but more rectangular (perhaps taking after the shape of the original astragali) or, in the case of some second millennium sites in the Southern Levant, "truncated four-sided pyramids".

The modern pattern came to be mainly in the Byzantine (~300-600 AD) and Islamic (600-1300 AD) periods, and it was during the Byzantine period and into the Islamic period that we begin to see dice with the modern arrangement of dots (see Wapnish, Paula. "Beauty and Utility in Bone. New Light on Bone Crafting." The Biblical archaeology review 17.4 (1991): 54-57.). To corroborate Paula Wapnish's article (she is easily one of the foremost experts on animal bones and bone crafting in the ancient Near East, and a fantastic person I count as a personal friend), I have excavated modern-arranged bone dice from these periods. But anecdotes are not considered evidence, so I am citing her article to back up my experiences.

If you are interested in continuing to read on the subject, as well as what games were played with the ancient dice and astragali, I suggest Schwartz, David G. Roll the Bones: The History of Gambling. New York: Gotham Books, 2006.

Supplementary question: what's the first known instance of loaded dice?

Another supplementary question: what's the history of the dice used in gaming, the D4/D8/D10/D12/D20s used in D&D and other games? Did Gary Gygax or Dave Arneson invent them, or were they already around?