There are a lot of rumors regarding US citizens opening offshore bank accounts and if it’s possible at all.
And while it's true that some European banking centers like Switzerland and Lichtenstein closed their doors to the US citizens’ investments, due to FATCA, it is and it’s always been perfectly legal to do business outside of the United States.
In fact, there are a lot of other, maybe unexpected places, where it’s quite easy to open a bank account, even for US citizens, like Hong Kong, Armenia, and Georgia.
All you have to do is to forget about the "historical" offshore havens and get the different mindset