Several players recommend a height about the thickness of a dime at the first fret, and about the thickness of a nickel at the seventh fret. The general strategy recommended is to bring the bridge and saddle a little bit at a time until you find a height that works for you.
How high should the action be on a mountain dulcimer?
It should extend about 3/32" above the surface of the fingerboard. Later, after the dulcimer is finished, this nut will be notched to space the strings correctly and hold them just barely above the height of the frets.
How do I buy a dulcimer?
Finding and buying a mountain dulcimer is a matter of looking and listening carefully: ask around, listen to various players. Look for a solid wood instrument; ask about its finish, request geared tuners, check its action, and sight down the top of the fingerboard to make sure its dead flat.