First you need to define "cheap" and "least expensive". If you don't mind building a cheap dulcimer, making your own can be fairly inexpensive - if you are already set up for woodworking. If you want a good dulcimer, e.g. nice looking, good tone, quality tuning pegs, etc, you will end up spending as much or more for supplies as you would buying one new.

And don't forget the tools you would need to build one - table or radial arm saw, band saw, coping saw, 12" planer, belt sander, drill and bits, possibly a router, bending iron, and a large array of clamps. Does all that fit into your budget too?

Oh yeah, and how many trials dulcimers do you want to build before you do the "real" one?

I've got 3 dulcimers of varying looks and sound quality that I made before throwing in the towel. And I had some experience in woodworking before taking on that project. The three now sit in a box in the garage.