The Mesopotamians didn't really use "currency" as such, which we can define as a centrally issued object having the function of store of account, medium of exchange and measure of value. In short, they did not have coins. But they did have standardized systems of weights and measures that, in practice can fulfill one or the other functions of currency. In Sumerian documents for example you will see value expressed in units of silver even if no silver is involved in the transaction. Currency in the form of coinage does not really come into being until the Iron Age in Anatolia, and it was taken up by the Persian Empire.