For Egypt, a number of tomb paintings and artifacts suggest that the Egyptians played a game called senet.

Unfortunately, no one knows the exact rules of senet. A number of archeologists and board game enthusiasts have tried to reconstruct the game based upon the surviving games and the paintings depicting Egyptians playing the game. Based upon these deductions, senet is somewhat similiar to backgammon in that one uses dice rolls (which could be bone dice or throw sticks ) to move one's pieces off the senet board. Gamecabinent.com has a good summary of senet and the two main rule variants (Bell and Kendall) and Brandeis University has a good writeup on how to play senet (warning- Brandeis link is a pdf).

In Mesopotamia, the Game of Twenty Squares, which was a kind of predecessor to modern backgammon, was invented just before 2600 BCE and is still played today. A cuneiform tablet from 177 BCE provides an overview of the rules that are likely to be very similar to, if not exactly the same as, the game's original rules set, but sometime during the early Christian Era the game either evolved or inspired the game of Game of 12 Lines, which was a similar, intermediary game that would later become backgammon. Additionally, thanks to the discovery of that 2nd century BCE tablet, the game was reborn in the 20th century and is played in numerous places all over the world today.

The game seems to have enjoyed wide popularity over thousands of years, and rough, ad-hoc boards have been found in Iraq, Syria, and even Egypt.

We don't have any more information about who played it or if it was esteemed or just casual entertainment, unfortunately.