The earliest coins known to archaeologists are electrum ones found at Ephesos (modern Efes on the western coast of Turkey) and dated to about 600 BC. These coins would have been struck in the kingdom of Lydia, confirming Herodotos' contention that it was the Lydians who invented coinage (Hdt. 1.94.1). The association of the Lydians with legendary wealth is preserved even in the English language: the expression "rich as Croesus" refers to the Lydian king Kroisos (595-546 BC), whose interactions with Greek elites and dedications at Greek sanctuaries gave the Greeks a glimpse of the riches of Asia. It was during his reign that the Greeks themselves first started minting coins, usually in silver. However, major Greek states like Athens do not appear to have had official currencies until the end of the 6th century BC.
That said, there were of course expressions of value before actual coinage, and these may be called currency. The oldest known from the Greek world is, simply, oxen; in the works of Homer, the value of objects, raw materials and slaves is expressed in oxen. Later evidence from Crete suggests the use of bowls as an indicator of value. Tripods and iron spits were also used in this way. They were not universal, but their value was easily understood even between communities, especially in the case of oxen. Similar systems would have been in place in communities beyond the Aegean before the introduction of coinage.