I love, love, love my bike.
The greenway on the West Side is the best (and as one poster said, it truly is the safest way to get from north to south), but some of the quieter streets in the West Village, Soho and Tribeca are easily bike-able too. If you stay aware (of both pedestrians and cars) and obey traffic laws, you will be fine.
My boyfriend rides his up Sixth Avenue to work every day and loves it. It's exhilarating and freeing.