That's an interesting question, but it is somewhat vague. We have to deal with two issues. (1) how to define money itself, and (2) how to define money laundering.

Not every culture throughout history has had the same ideas about money. But let's set that aside for the moment. Money Laundering typically refers to the process of turning the proceeds of an illegal act into legitimate (or at least apparently legitimate) money.

First off, the term "money laundering" itself didn't arise until the 1970's. It was first used to describe already existing processes when federal legislatin in the US was passed to ban those types of activities. However, the concepts existed far earlier.

If we use a broader definition of money laundering as any act taken to hide assets from the government, the practice dates back into antiquity. Ancient Chinese traders/businessmen in the far east worried about rulers who could, if they desired, seize their assets and banish them from the country, so they developed practices to "offshore" (in some cases literally) their assets, and protect them from political corruption. Sterling Seagraves discusses this to some extent in The Lords of the Rim.

another early example might include mediterranian pirates who figuratively "laundered" gold and other precious metals seized from merchants and traders. This would have been accomplished merely by melting down the stolen metal into bars and re-creating other currency, thereby disguising its source. Once a ship pulls up to harbor, gold or silver bars are fungible and can be traded without any difficulty.

However, if we use a more technical definition of money laundering to mean a system to disguise the source of illegally obtained cash, this almost certainly dates to the modern era. To figure out why we have to look at the history of money and banking to some extent.

The idea of "paper money" arose quite a long time ago in the form of letters of credit or various other notes. However, paper money (Bank notes) only became widely used in the 18th and 19th centuries and paralleled the rise of the banking system. Money laundering in a technical sense didn't become necessary until the widespread use of banknotes. If the only currency that exists is gold or silver coinage, as noted above, it can simply be melted to disguise its source, then sold for equivalent value.

The usual cited place for the origin of money laundering was the US in the 1920's, when prohibition led to large scale smuggling of alcohol and organized crime. When combined with a modern economy and banking system, this led to a new problem. A smuggler has a cash business, and therefore accumulates cash income, but can't spend that cash income without incurring suspicion by the authorities. He has to create a fictitious source for this cash to avoid incurring the suspicion of the authorities.

The usual method of doing this is to create a ficticious business front in the community. Typically one that would operate in cash. This business reports much higher income than it actually gets, and then its "profits" are legitimately taken out of the company. Illegally obtained cash becomes the legitimate profits of an "established" company. A "laundry" could have been one such source, but what I've read suggests that this is actually an urban legend, laundries were never or rarely used for this purpose. Restaurants, gambling establishments and the like are far more likely sources.