Per Jack's comment, you can play slide guitar, either in regular position or with the instrument on its back. Usually slide is played with the guitar in open tunings, commonly open-G (DGDGBD) or open-D (DADF#AD).
I'm not saying it's better than a dulcimer, but:
- you've got more range of pitch on a guitar;
- with multiple strings you can play true chords if you want;
- dulcimers are mostly very quiet, so you often won't be audible if you're playing with other people;
- a dulcimer requires *more* force to play than a slide guitar;
- you can play slide in standard guitar position, whereas dulcimer requires some relearning of left arm movements and positions;
- you already know guitar, and open tuning isn't much of an issue;
- you already have a guitar, so it'll only cost you the price of a slide to try it.