TheTokenBrit, when we lived in Austin, the husband used to get treated all the time with free stuff all because of his accent. He had been given extra donuts at the donut shop, free dry cleaning, bus passes and a few free drinks in restaurants. Now, he works as a sales engineer and I hear how people respond to him because we both work from home and share an office. He has managed to secure sales and smooth over a few prickly customers because they enjoy chatting to him and it sort of breaks the ice and reduces hostility and many times they ask directly for him. Like Sonofasailor mentioned, it does have a sort of cachet.
To be honest, I don't hear it anymore and it surprises me when someone points it out. His family keep commenting that he speaks with a slight American accent and I definitely don't hear that!
I've been told time and time again that I have a neutral American accent but thanks to the 22 years with my Brit, I confuse people a little bit because I am half-American and half-Mexican and a Texan who uses English/ British/ Mexican words and expressions in every day speech. But I think it's rubbing off on the guy. He recounted a story to me about how he went to Home Depot for cables: "Can you imagine that there were three cables exactly the same but with three different prices of $5, $6.50 and a whopping $22. Yes, $22! I was thinking, 'Están locos, $22! Are they made of gold?' Bless their hearts that they think we are a special kind of stupid". Lol