The three-or-four string "Appalachian dulcimer," "mountain dulcimer," "lap dulcimer," or "plucked" dulcimer, while not unique to Kentucky, has long been associated with the highlands/mountains in Kentucky and other Appalachian states. Cf. the historical overview online.
A completely different instrument, the "Hammered Dulcimer" -- a forerunner of the piano -- is not associated with Kentucky, and is often confused with the "Appalachian dulcimer" because of the accidental similarity of its name.
One of the most celebrated modern Kentucky plucked dulcimer builders was Winchester, KY native Homer Ledford (1927-2006), who built nearly 6000 dulcimers during his long career as a luthier. A short highlight overview of Homer Ledford's career is on Wikipedia, under "Homer Ledford."
KENTUCKY DULCIMERS MUSEUM/LIBRARY COLLECTIONS
KENTUCKY DULCIMER PLAYERS and ORGANIZATIONS