There's a funny article in the NY Times about how many linguists believe Mayor Bloomberg has shed his Boston accent for a... well, it's hard to say whether or not he speaks New Yorkese, but it's definitely less Boston than before. There is a funny quote from Barry Popik, the administrative judge, former candidate for Manhattan Borough President, member of the American Dialect Society and Big Apple enthusiast:
"It appears to me that Bloomberg's been studying two new languages - Spanish and New York Jewish. He's not sounding like Fran Drescher's 'The Nanny' yet, but it appears to me that he's picked some of that up."
General consensus is that it sounds like Mayor Bloomberg is making an effort to sound every-New York-man, and while that's great, Gothamist just wants the video footage of the Mayor singing, "The rain in Spain..."
The linguists used his inauguration speechs from 2002 and 2006 has comparison, and there's a produced online component where you can hear clips and analysis.