A lot is explained when I read this quote... Mayberry (overlooking the modern cars and so on) was really set in the 1930s, as far as characters and storylines. In a way, perhaps based on Andy Griffith's childhood in Mount Airy than is commonly recognized.

Regarding the time-period of the show, Griffith said in a Today Show interview:

"Well, though we never said it, and though it was shot in the 1960s, it had a feeling of the 1930s. It was, when we were doing it, of a time gone by."