Here is a very nice simple traditional song- All the Good Times Are Past and Gone. It's especially good for beginners (are you with me, "papabill"?). I've presented it in the key of G, because that's the key I usually see it written out in. Also because you don't want to get stuck in the rut of only being able to play in the key of D... I've tabbed it for G 'reverse' ionian tuning (DGD). I presented a chart a few posts ago showing how to make the change from D 'regular' ionian into G 'reverse' ionian. Using the reverse ionian tuning sometimes can make the re-tuning process a little easier by changing the strings a bit less than going from regular D ionian to regular G ionian. I know that statement might be confusing- I am going to explain that a little further one day very soon. But do notice again that the song ends on fret 3, which gives us a hint that it's in ionian mode.
I know you probably are loathe to retune your dulcimer into the key of G from your 'comfort area' of being in D...but getting over the natural tendency to want to avoid re-tuning is going to allow you much greater freedom to play all kinds of songs and tunes later, and will give you the ability to play in fun jams of other music playing people... people who play in several different keys on other instruments. It's only hard and scary to change tunings the first few times you do it. After that it becomes much less of a big deal.
By the way- did you know the difference between a 'tune' and a 'song'? A tune is a piece of music usually with a melody, and a song is the same thing but with words/lyrics added to sing it. You sing a song, you play a tune. If you play a tune and then add words to it, it becomes a song!
"All the Good Times..." is a very good song to try to singing to while you play, since the first verse is pretty easy to memorize and sing on 'autopilot' while you play. Being able to sing a song while you play it is an immensely helpful skill to practice. It doesn't matter if you don't sing well at all, you can just sing by yourself all alone if you prefer. Having a slightly crude and rustic voice is actually something that's admired in old-time music! You'd sound pretty lame if you sang these old country songs in a trained operatic voice.
Practicing singing something simple while playing actually helps cement the song into your brain so that you can better play it without tab, and it helps you to keep rhythm better. Think of how easy it is to remember little children's songs from your youth- this is much the same thing. Plus, there may well come a time as you get to be a better player when you will actually want to sing songs with others, and if you practice this here and there during your early learning stages it won't be so hard to pick up the skill later.
As will all things, my advice is the same- start simple and slow, and don't get more complex until you feel comfortable. Remember this: it's far better to play 3 simple slow tunes well than to stumble awkwardly through 8 fancier tunes.
In this case, I'd suggest singing just the first few words with your dulcimer playing over and over: 'all the good times have past and gone'. Once you can play and sing that phrase comfortably, add the 'all the good times are o'er' to it. And so on. If you never get past the first verse, that's fine too!- at least you have shown yourself that you can do it.
Here is the song, in the key of G, ionian mode (but remember you can play the tab exactly the same way if you are tuned to D ionian tuning, DAA, as well):