We have known how to do untraceable transactions since 3,000 BC.
1. When you get cash from an ATM or Bank they record the serial numbers of the cash they gave you and attach it to your identity.
2. When you spend those funds at a store they record what you bought and the serial numbers of the cash you paid with and the serial numbers of the bills you received as change.
The government could then build a graph of spends attached to your identity. With high probability they could trace how you spent that money. One problem they would run into is if you gave or received cash from a friend. However while this might cause the tracing procedure to make some errors, it would also allow it to, in some cases, infer which of your friends you exchange cash with.
In any realistic scenario the government would also be intersecting this graph of spends with additional information such as license plate readers, RFID data from public transit, facial recognition,cell tower geolocation, etc... Serial number tracing would be just one data source.
Probably your best bet for privacy here would be pay people begging for money to swap bills with you e.g. exchanging a twenty dollar bill for a ten dollar bill.
[0]: https://en.wikipedia.org/wiki/Ecash
* - For some values of untracable. The privacy provided by ecash is in theory less than cash provides in practice, but certainly a massive privacy improvement over credit cards. In theory cash provides less privacy than ecash since stores could record the serial numbers on cash. However in practice they don't and thus cash in practice provides quite a bit of privacy.