I play this in what I call a "broken drone" style. I suspend the drone and pluck the melody at certain points as shown in the tablature below.

The song is in the key of G and the dulcimer is tuned DGdd (sometimes called "new ionian"). This is an easy tuning to reach from the popular DAdd – just tune the middle string down one full step. The exact same fingering can be used to play Amazing Grace when tuned DAdd but you'll find that the DGdd tuning is "sweeter" with the drones.

Finally, my personal style on this song is to lose the pick and brush the strings toward me using the ball of my index finger. This gives a much softer sound and also allows for a lot of expression.