I'd say it depends who you are and comes down to 'what's stopping the US carrying their threat out'. That splits nations down roughly into these groups:
Member of NATO: not really applicable. The U.S. would risk the rest of the alliance turning on it and, although not as strong individually, the non-US NATO countries together would still pack a hefty punch. Besides which, the NATO countries are America's closest allies, so they are unlikely to be threatened, never mind actually attacked. Also applies to a lesser extent to other U.S. allies.
Other nuclear states: India, for one, but mainly Russia and China... These are too much of a threat if they responded using nuclear weapons. Again, the U.S. isn't going to seriously attack them during anything less than a global breakdown, so there's little use in trying to idly use a threat of force. They wouldn't care particularly; they know it's hugely unlikely it would be carried out.
Friends of the above... North Korea, Iran predominantly, but note that even in Syria the U.S. is only attacking ISIS, not the government. They're probably more concerned about losing the support of their patron than the threat itself, as they are sheltered under the umbrella of the above states.
Anyone else? I'd say they take it fairly seriously: the US has proven in the past it isn't afraid to intervene when it feels its national interests or security are at play, and if you aren't in the above listed groups you are very unlikely to be able to defend yourself effectively. In fact, without nuclear weapons, I doubt anyone other than the rest of NATO as a bloc could seriously hope to prevent the US from doing what they wanted militarily. Russia and China would be the only other ones with a fighting chance.
As such, any nation outside those groups would likely take the threat very seriously, although they also know that public pressure in America has a large impact and so would weigh the threat against the reasons it was made.
