The Gaels never had their own money, even well into the 'Middle Ages'. The Incans, likewise, built a complex civilisation without a role for money.
I suspect a lot of the acephalous and indigenous societies of Africa function/functioned without money too. The Maasai I know go without it by custom. In Tibet before 1650 it's debated.
Did the pre-Columbian North Americans have any money/coins/gold? Australian aborigines had none I presume? Pacific island cultures?
Has anyone done a statistical survey of what cultures have the stuff or don't? I'm asking because I suspect that there's a paradigmatic expectation for societies to have money, and so we go looking for the ancient coins, gold, silver, but I suspect it's absent in many cultures.
/u/yemrot has previously answered Where did the idea of currency came from? Why many different nations from different side of the world will have the similar idea of using a common "something" to replace the barter system? How did nations with different currency trade with each other? but someone else can write an answer to your specific question.
EDIT: /u/drpeppero and /u/savageson79 have previously answered How did the Inca Empire function without money?, among others.