It's easy to refute, but you end up knocking down the same arguments, over and over. For example:
The Democrats are the Party of the KlanNow, it's true that Nathan Bedford Forrest was a Democrat even before he set up the first iteration of the Ku Klux Klan (there have been three, if you're curious). And it's also true that Abraham Lincoln was a Republican when he was first elected.
(Fun fact: Lincoln left the Republican Party at the end of his first term. Republicans are rarely aware of that: for his second term, Lincoln created the National Union Party, a coalition party made up of both Republicans and Democrats.)
But here's the thing: at that point in time, the Republican Party was liberal, and the Democratic Party was conservative. (This fact particularly angers the Teabaggers, who've been brainwashed to think that liberalism is synonymous with "evil.") And from the Civil War to about 1950, the Southern Democrats (sometimes called "Dixiecrats") were among the most conservative (and usually racist) people in America.
The Dixiecrats never ran another presidential candidate, and eventually the party dissolved. And following that victory, the liberal Democrats became a stronger and stronger force in the party, eventually reversing the formerly conservative platforms, and passed the Civil Rights Act of 1964 and the Voting Rights Act of 1965.
This same action, of course, drew the Republican Party to the right, in an effort to pick up the disillusioned Southern Democrats.
Following the 1964 Civil Rights act, LBJ famously said “I think we just lost the South,” which would prove to be remarkably prescient: in the late Sixties, Richard Nixon, with the help of his advisor Pat Buchanan, devised the "Southern Strategy," using dog-whistle racist terms (example: "states' rights" - the states would have the "right" to ignore these new civil rights laws).
In 1980, Ronald Reagan (working with Nixon's advisor Pat Buchanan) further honed the "Southern Strategy." In fact, it was another of his aides, Lee Atwater, who famously spilled the beans years later, thinking he was speaking off the record to a reporter.
You start out in 1954 by saying, "Nigger, nigger, nigger." By 1968 you can't say "nigger" — that hurts you. Backfires. So you say stuff like forced busing, states' rights and all that stuff. You're getting so abstract now [that] you're talking about cutting taxes, and all these things you're talking about are totally economic things and a byproduct of them is [that] blacks get hurt worse than whites. And subconsciously maybe that is part of it. I'm not saying that. But I'm saying that if it is getting that abstract, and that coded, that we are doing away with the racial problem one way or the other. You follow me — because obviously sitting around saying, "We want to cut this," is much more abstract than even the busing thing, and a hell of a lot more abstract than "Nigger, nigger."And that's where the Republican myth of the "racist Democrats" comes from: the Dixiecrats, and the changing face of the Democratic party. Back when Democrats were the conservative party, they were, in fact, racist; in swinging to the left, they also became the party of racial equality. To the point that, yes, the Ku Klux Klan may have been founded by Democrats, but these days, while not every Republican is in the KKK, almost every Klansman votes Republican.
Edit: (12/6/14) Corrected "North Carolina" to "South Carolina," with apologies to any North Carolinian in the audience.