Thank you all for your replies. For some songs, I agree that the percussive click adds a cool rhythm element, just like it does on guitar. It made a really big difference when I switched from the thin, large triangular pick that came with the dulcimer to one of my thicker guitar picks, where I don't have as much of the point exposed. As a new dulcimist, I wouldn't have reached those conclusions on my own this quickly without your help!
Last night, I realized I'd learned all but one chord to James Taylor's cowboy lullabye, "Sweet Baby James," so I looked up Bm and started playing. Suddenly, it felt like I was back on the steps at my grandparents' home in south Arkansas in 1972, struggling to change through the same chords on my first guitar. So my new-to-me dulcimer has already brought back some wonderful memories and enabled me to appreciate the joy of learning an old, old favorite--all over again in a new way.