Ike was a solid moderate, JFK was a moderate liberal. The worst myth I've heard is the "conservative Obama" one.
The JFK one. People who were Democrats but became Republicans in the 1964-1980 timeframe often say that everything was great under JFK and that his assasination was the beginning of America's decline. Whether Eisenhower was conservative or liberal isn't relevant to anything that happened after his Presidency.
Most people who were Democrats before 1964 didn't become Republicans until after 1980, or were neocons, not Dixiecrats.