The only time having a heavy accent is bad is when a person gets pulled over by the cops. Heavy Southern, NY, NJ, Philly, Boston, BMore accents will all instantly make the police judge, unless the officer has the same accent. It can be remedied quickly by injecting large words into the conversation. That's the advantage a Southern accent has over the others; an educated Southern accent is inherently less threatening to police than an educated NY, Philly et cetera accent. Police will also be instantly defensive and suspicious of foreign accents.
Personally, the physical sound of the voice has a great impact than the accent. If someone has a pleasant voice the accent generally does not matter. If someone has a grating or whiny voice, then their enunciation had better be on point.