Well, no way to "legally avoid"; "dodge" is broader, and there are probably many ways of dodging taxes that are easier if the institutions you deal with in your daily life aren't subject to US jurisdiction and regulations designed to expose tax evasion to US authorities.
Also, typically you want to have at least one national citizenship, so if you are planning to legally avoid US taxes by renouncing US citizenship, getting some other citizenship is usually step one.
Of course, there are other reasons it is convenient to have citizenship in an EU member state, even if you are a US citizen, so its not necessarily step 1 of a tax dodge.