For a pleasant harmony, try fretting the middle string two frets behind the note fretted on the melody string.
Or the bass string one fret behind the note fretted on the melody string.
Or all three together (e.g fret the melody string at 3rd fret with index finger; middle string at 1st fret with ring finger; bass string at 2nd fret with middle finger).

If you've got a chopstick handy, I bet "Oranges and Lemons" would sound great on the old "imitation hammered dulcimer." Fret with your left hand as you're doing now, but instead of strumming with right hand, tap out the tune with the chopstick -- don't hammer it; cock your arm up so that your hand is a few inches above the fret board and allow the chopstick to drop onto the strings in the appropriate rhythm.