A deserter from the US Army, who crossed into North Korea and got citizenship, was court-marshalled as he arrived in Japan with his Japanese wife, whom he married while she was held captive there as an abductee. Though the court, meeting at Camp Zama, was mainly dealing with his apparent desertion, his acquisition (involuntary?) of North Korean citizenship was considered as part of anti-US activities (which included appearing in an anti-US propaganda film). He was dishonorably discharged and served a short term in prison before being released. He had to receive a special short-term visa to visit his ailing mother in the States. He has since renounced North Koren citizenship and applied for the return of his US passport. While it may be argued that North Korean citizenship isn't just another nationality, given the lack of diplomatic relations between the two countries, it's incorrect to state that a voluntary acquisition of another passport has no consequence in every case.
I'll just point out that the OP has revisited TT several times since s/he posted this. I somehow get the impression that his/her interest in this thread is now non-existent since walkintheworld hasn't even acknowledged these replies.

#10 I didn't state that it had no consequence in every case, I stated that it had no consequence absent extraordinary circumstances. As far as I can tell OP isn't considering defecting from the US army in favour of a country the US has recently been at war with (which seems to me to be the real issue in Jenkins's case not his acquisition of a North Korean passport) so it's not really much of an argument.
Just to point out also in #7 you refer to sanctions for people "breaking rules". OP would be breaking no rules by acquiring a European passport.
A fact to bear in mind, in all this, is that US has moved in its position regarding dual and multiple citizenship by US nationals from active hostility/discouragement to neutral. The federal government has no opinion on people who acquire another citizenship, and views the giving up of US citizenship, often required under another country's rules prior to naturalisation, as having no legal effect, unless it is accompanied by proper procedure in front of a US government or diplomatic official. It does require multi nationals to use US passport when exiting and entering the States, and as is common under international law, it cannot offer protection or assistance to nationals who happen to fall foul of the law in a country where they are also citizens of. Also, while somewhat unrelated to nationality rules, US citizens including dual nationals are subject to US federal taxes and are required to complete Internal Revenue tax return each year, whether they are deemed US residents or not. They don't however have tax liability in every case, as there is an annual exemption (around $86,000 in 2009 plus $14,000 in housing allowance) as well as dual taxation relief etc. In many other countries, once you establish ordinary residence for tax purpose in another country, you are no longer subject to income tax in your home country.
This stance, while an improvement on previous position, is a long way from quite liberal rules that exist in UK or Ireland, which put no obstacle or requirement in the way of their citizens acquiring another nationality, including using native passport for leaving or entering their respective country.

My wife (US citizen) got her irish passport ....... no renouncing of any sort, and I also doubt the US government even know she now has one.
#14
As I've said Ireland like UK have very liberal multi-nationality law and allow other citizens to acquire Irish nationality and Irish citizens to get other citizenship without restriction or pre-conditions. A lot of other countries put a number conditions, and renunciation of existing nationality, except in limited circumstances, is quite common.