There is no single "Southern accent".  Instead, there are a number of regional dialects found across the South, collectively known as Southern American English.  Yet these dialects share features of accent and idiom that distinguish them from the English spoken in other regions of the United States.

If you draw a line roughly from Texarkana to Jonesboro in the north half of the state, the dialect is classified as South Midlands, with a spot of Ozark along the Missouri line, and the dialect of the southern half of the state is called Gulf Southern. The line separates the cotton growing south from the smaller land holders of the north. The general southern dialect has its origins in the immigrants who moved to the South in the 17th and 18th centuries.  Most Southerners have surnames that are English, Scottish, Irish or Welsh in origin. These immigrants brought with them a very distinct style of English speaking which was combined in the South with the west African languages spoken by slaves.  Over time this cultural and linguistic diversity combined with the South's rural isolation, and longtime use and familiarity with the King James version of the Bible, to produce a unique American dialect; the Gulf Southern.  These features are characteristic of both older Southern American English, which underwent changes during Reconstruction and is now largely rural and rarely used by those born after World War II, especially with the advent of television in the 1950s.   Traditional southern grammar included such uses as:
*  fixin' to as an indicator of future action.  For example: "He's fixin' to eat", or "We're a-fixin' to go".
*  Use of double modals (might could, might should, might would, used to could, etc.) and sometimes even triple modals that involve oughta or a double modal (like might should oughta, used to could be able to, etc.)
*  Addition of adverbs here or there after this or that;  "Johnny, fetch me that there hammer."  This is mostly used in rural speech.
*  Use of done instead of have in perfect constructions (perfective done), as in, "He done come up here", or, "I done told you."
*  Replacement of have (to possess) with got, as in, "I got one of them."
*  Using them as a demonstrative adjective replacing those.  "See them birds?"
*  Use of irregular preterits, such as drownded as the past tense of drown, knowed as past tense of know, degradated as the past tense of degrade, and seen replacing saw as past tense of see.  This also includes using was for were, or in other words regularizing the past tense of be to was.  "You was sittin' on that chair."
*  Use of unmarked verb preterits; "They come in here last night."

But the letter 'R' is an entirely different matter.  Older Blacks tend to pronounce it, "ah ra", country folks might say for the word 'where', "whahra".  The educated pronounced 'R" as "ah", and added an extra syllable which was, as the French say, pronounced but not heard.  My Mississippi-born father lived in Arkansas for almost eighty years and was kidded to the end of his life for his lack of an 'R' sound.  "Rabbit" came out "abbit" and my mother accused him of having a speech impediment.

Like anyone learning a foreign language, African slaves substituted sound patterns from their native language for unfamiliar  patterns in the new language. Most West African languages end each syllable with a vowel. English words ending in two consonants would be difficult to pronounce, so the slaves who were learning English would have a tendency to drop the "r" at the end of a word or before a consonant.  Over time, as the slaves and the white planters were in daily contact, their speech patterns evolved along similar lines, except the planters were exposed to many outside influences, while Black culture at that time was insular.

Thus in the middle dialect of bondsman and planter both dropped the "r", but the small land holding settlers, who had little contact with bondsmen, or with planters for that matter, retained the "r". These white country types, often poor and uneducated,  might say "whar" for where and "far" for fire. Later, due to the long standing  competition between the poor whites and the poor blacks for the same jobs, the hard "r" from a white was often taken as an indication of a racist. 

Those growing up in town did look down on poor white country folks. I remember my country school mate saying, much to my amazement, "Shit far, the teacher done rung the school bell."

The three estates, two without the "r" and one with the "r" formed a kaleidoscope through which historians and writers have seen the post antebellum South. In Faulkner, the aristocratic Sartorises are replaced in the power structure by the pernicious Snopes family.  In the novel, "To Kill a Mockingbird," Atticus (read Sartoris) answers Jem's question of how a jury (composed of members of the Snopes family) could convict innocent Tom Robinson (the victim of racism) by saying,

"They've done it before and they did it tonight and they'll do it again and when they do - it seems only children weep."

Well, all of that has changed now. Nobody, as the media point out, wants to "play the race card."  The descendents of the Snopes family who might still feel threatened by a society committed to diversity can only react by joining the Tea Party movement.
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Many speakers around Penhook are bi-dialectic.  When my Kansas-born wife first visited Penhook we were driving down a dirt road and I encountered a black man I had known from childhood.  We stopped our respective trucks, rolled down the windows and visited for a while.  After we parted ways I noticed that my wife was very quiet.  I asked her if she had understood what the man had said.  She replied, "I didn't even understand what you said."

If there is a sort of shared southern dialect where blacks and whites comfortably converse there is also a unique black dialect.  Once I was in the liquor store near the Red River and two black men were having an animated conversation.  I noticed the white clerk was paying attention to them.  I could understand only part of what they were saying. After the men left the store I asked her if she could understand what they were saying.  "Yes", she replied, "but I wish I couldn't."

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There is more to language than just grammar. Southerners delight in the use of language.  In the early 1950s when I worked behind the meat counter of a grocery store at the county seat, on some Saturday nights we had a special visitor.  Around 9:00 pm when we might be cleaning out the meat cases and getting ready to close, a great bass-baritone voice might be heard.
Jonas Jones, a black preacher with snow white hair, would sing out, as he strode down the grocery aisle, "Hitherto shalt thou come, but no further; and here shall thy proud waves be staid." He would pause, then conclude, " Job, Book 38, Verse 11". Then he would stop just in front of the meat case and emit an infectious laugh that filled the store. 

Jewell, the owner of the store, would reach over the meat counter and shake Mr. Jones' hand.  As they were old friends a long conversation would follow and the boys went back to cleaning the meat cases.

There was another great voice that quoted Job, a black share cropper, whose name a don't remember, would be hoeing  his cotton when the grass was about to take over the entire patch. A  mile away he could be heard singing  out, "Gooood Almighty Damn, Man born of women is of few days and many troubles!  He blooms in the morning like a flower and withers in the noon day sun!"
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Sam Williams reports this  marriage contract drawn up in the 1830's by a backwoods couple.

We may smile at the quaint phrasing, but compare the marriage contract to a modern example written by someone with, I am sure, several advanced degrees: