Well, well, another thread which should generate some great responses! I learn by ear a lot of what I play. Indeed, I feel more comfortable more quickly when I learn by ear. However, you can make mistakes by mishearing or misremembering some of the notes. That's where you talk about "regional differences" in tunes (works particularly well for fiddle tunes not so much for a Johnny Cash song.) Training your ear is, like Robin said, not difficult and well worth it. Start with a simple tune you know well but have never played on the dulcimer; then find the notes. In fact, if you like Johnny Cash a lot of his songs meet the criteria of "simple tune."
The next best way for ME is SMN. I just learned O'Carolan's "Lord Inchiquin" from SMN in G and played in D. I think Robin's mention of intervals brings up a great thing which happens (I think) to at least many dulcimists who read music: you quit thinking in terms of note (a, b, g#) rather you think about the interval between the notes (1, 2, 7.) Then it's easy to transfer the SMN, no matter what key it's in, the dulcimer, no matter what tuning it's in.
Lastly, I hate tab. I hated it on the guitar; I hated it on the banjo; I hate it on the dulcimer. But.... I use it when that's all that's available. Now, I don't mind tab below the SMN. I think of that as the fingering indications on classical guitar music: read the note, get an idea of where it should be played on the fingerboard.
There you go. My 2 centavos! Have fun
Rob