Other questions on this site have established that kindly can be used as a sort of please. This usage was in my mind when someone said "Thank you kindly" to me, but "thank you please" doesn't make sense. However, characterizing one's own thanks as kind also sounds unlikely. The actual usage appears to be the sixth definition by Dictionary.com:
kindly
6. cordially or heartily: We thank you kindly.
However, I still have questions. Is this usage of kindly only used with thank you? How did this usage occur?
Best Answer
Where I live (SE USA), kindly still is very much in use, but mainly as a way to exaggerate politeness, sometimes tongue in cheek. The word does not literally mean the same thing in thank you kindly as it does in would you kindly?, but in both cases it carries the connotation of being so polite that it sounds a bit silly, which makes it polite again. I realize that makes no sense whatsoever but I suppose it's a Southern thing and ask that you kindly make allowances.