According to Aflac's website, the average claim turnaround time is less than four days. This would include all policies with a "paid" status of all types, including dental, cancer, accident and life.
Recently, Aflac introduced a program called "One Day Pay," purporting to help pay claims faster than ever before. This hasn't changed Aflac's already low turnaround time, but they are hopeful it will change their business and the industry.
Note that for a claim to be paid out, it must be submitted correctly. Aflac would not include claims that were not submitted correctly in their assertion that they pay in under four days. Aflac also notes that if a claim is filed online, the claimant benefits from quicker processing time - the claim gets there instantly.