My experience has been that a dulcimer builder who knows the woods they are using, knows how to test their vibrating characteristics, can definitely and specifically influence the voice of the instrument.
In fact, after doing so for a number of years and keeping careful notes of choices and the resulting voice, a builder can forecast what the instrument will sound like, and thus custom shape the voice for the client.
This all depends, of course, on having enough stock of each wood used so that one has a range of choices. And that the woods used are adequate for good voice quality.
And that depends on what the client wants from the dulcimer. That's why so many different dulcimer builders are at work: there are so many differences in what a musician is looking for in a dulcimer. In an instrument like the violin, there is such a long history of what works and what doesn't, there is far less variation in what a violin should sound like.

My experience is that there are a huge number of variables which affect the sound of a dulcimer, and that the kind of wood is pretty far down the list. A competent luthier can make almost any wood (or even cardboard, plastic or Legos) sound the way you want by adjusting any of a number of variables such as bracing, cubic sound box area, VSL, break angle of strings, details of fretboard construction, etc., etc.
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