I know it is relatively common in parts of Africa due to how common counterfeit bills made of the older series are, especially central and eastern Africa, as well as SE Asia. We fell afoul of it in Kenya with bills printed before 2007 (I knew beforehand but hadn't checked my currency carefully enough), however we had no issues changed the notes in South Africa.
Places it has been noted as an issue:
However in some of the countries, mainly the non-African countries, it can be a site by site policy rather than a generally accepted one across the board. For example, in Australia I suspect (as the answer suggests) a bank would exchange the notes, instead of visiting a currency exchange.
My recommendation for travel outside of Europe/North America would be to exchange your older US dollars in your home country if they are accepted there, either for fresh notes or something else that's generally convertible like euros or pounds.