I use the most common Mixolydian modal tuning, DAD (low to high, with the high D as a unison pair). Lends itself best to adapting the widest variety of songs, especially Celtic and those played in open tunings on guitar. I play mostly two-finger chords with either the bass or the treble pair as drones, and I play melody notes on all three strings. (This was the method championed back in the mid-70s by Force & D'Ossché in "Stalking the Wild Dulcimer," where they advocated holding and strumming the dulcimer like a guitar). Traditional players originally used Ionian, or DAA, playing the melody usually only on the treble pair (sometimes a single treble string), with a single finger or a stick called a "noter." David Massengill uses a "reverse Mixolydian," or ADD (or since he prefers to sing in C, CGG, with a single treble string). He usually chords using the bass and treble, with the middle as a drone, except for more complex chords. He plays his melodies using mostly chords rather than single notes. Janita Baker pioneered the use of equidistant 4-string Mixolydian (D-A-D-D) as well as variations, in order to play more complex jazz & blues chords & scales.
Those are primarily major modes, which also allow you to play in the relative minor keys and the keys a fourth above the bass string. Dorian mode is the most common minor mode. www.everythingdulcimer.com and www.fotw.com have much more info as to various tunings.