What is the origin of the accents in American English speakers?

A manner of speaking for a specific country is called an ACCENT, such as a British Accent, or a an Italian Accent. Within a specific country, there are regional DIALECTS, “such as the Southern drawl, Boston twang, [and] the California sing-song,” that you mention.

One’s accent is usually learned by the people surrounding a person when the person is in his/her formative years, such as a toddler who is just learning to speak, and s/he literally mimics or imitates whatever accent and/or dialect s/he hears.

This would explain why the late bandleader, Lawrence Welk, had a distinct German accent. He was born in Strasburg, North Dakota (USA) in 1903, but the community into which he was born was an exclusive German settlement, and the residents there spoke German, exclusively. Therefore, as he noted in his autobiography, “I didn’t learn English, until I was 21 years old.” Please note that some sources refute that, but I’m quoting him from his own writings.

Source: http://en.wikipedia.org/wiki/Lawrence_Welk