By extension of the telly, the world is a smaller place giving exposure to countless dialects. However, being exposed to them in person is a different matter. I personally am a Glaswegian with 20 years of living in London to dilute my brogue. The value of it had never really occured to me until a recent visit to New York. I thought my accent was mild bordering on imperceptible but that didn't stop me ending up in a bar in Greenwich Village surrounded by a comical group of female interest asking me to keep saying phrases over and over again. I guess there must be something to it...
For a Brit in a place like New York it's too easy to be consumed by the pace of the local tongue - if you choose to maintain your own rhythm and inflection then I would suggest that has as much effect as the finer points of pronunciation.