A 158 tuning such as DAd will give you all the chords that BELONG to the key that your 1 indicates, but chords from outside of that key will not all be possible. On a diatonic dulcimer, a certain amount of tuning changes is normal to accomodate different keys and/ or modes. A chromatic fretboard will give you all the notes, but some of the quick simplicity of the diatonic fret pattern is lost, so there is a somewhat longer learning period. Not so different from mandolin, guitar, etc, but there is more to deal with. No single three string tuning will compensate for all the missing frets. If the music you intend to focus on requires frequent use of chords that come from outside the basic key, a four string tuning as mentioned above is probably your best bet. 

Angela -- rather than going to The Dark Side, with a guitar teacher, you should be able to find a Hummel (or is it Homle) teacher there in Holland.  The Humle (or is it Hommel) is the Dutch equivalent of the mountain dulcimer -- a fretted zither -- usually played on the table top.  Same tunings as the dulcimer, perhaps more drone strings, but played the same.  The Dutch instrument is not chorded but rather played either noter & drone or fingerdanced. 

I was watching a Bing demo and he mention jamkazam        https://www.jamkazam.com/          not sure what it's like but sounded like maybe you could get with a dulcimer group for jams over the internet.

Also, Bing has many lessons on video called Dulcimerica , he just did his 300 -  all free and you can learn on your time. Here is one on fingerpicking and off to the right you will see many others to choose from. Check them out and have fun:  

Stephen Seifert has some free ones and like many others they do offer lessons on line, so you could learn but not with a guitar teacher, or with  both

Like Travis, I learned melody first on MD and chords on uke and guitar.  True story: some years ago when I had just figured out how to chord on MD, I was jaming with a uke-playing friend.  I was frantically grasping at the chords (would have helped if I knew the song better).  We finished singing the first verse and chorus and I thought thank goodness, now I can take the instrumental break and I started playing melody which was so much easier.  And my friend said, "oh that's right, unlike me you really know how to play!"  The easy beginner stuff for me was the hard stuff for him, and vice versa.  

I take this as proof that we're all meant to play together. 

I don't necessarily have an opinion about whether one should learn melody or chords first, but I can corroborate Travis's assumption that most guitar teachers start with chords.  I think the reason is simply that with a few chords one can quickly accompany one's singing. It is also the case that most single note work on the guitar is built out of left-hand chord positions (I am not joking that it took me 15-20 years of playing to figure that out, but I am almost entirely self-taught).

I certainly wish you the best, but I do agree that a guitar player who is not familiar with the dulcimer may not be the best teacher. You might consider Skype lessons from a dulcimer teacher.  Just the fact that you are asking about rare chords makes me think your teacher may not really "get" a diatonic fretboard.

Melody, melody, melody.   If you don't have the melody correct there is no need to add chords.  Were I to teach a beginner on either guitar or dulcimer scales would be the 1st thing I'd cover.  I know a fellow here in WV who carries 7 Jim Good dulcimers in a cabinet tuned in 7 different keys.  I agree with Travis, you have to approach them differently.

Angela--  No dulcimer tuning really has a wide range.  That's part of it being a dulcimer, not a guitar -- it is not chromatic.  If you want to play a lot of minor chord music, you might be best off having two dulcimers -- one tuned in a minor tuning like DAG or DAC; the other tuned in a major tuning like DAA or DAd.  Otherwise re-tuning is your best option.  Usually you only have to re-tune the melody string....

Thanks for bringing that up  Dusty. I didn't think to mention that, I was on a tablet and a bit pressed for time.

I've also tried DF#A and DF#AA [4 equidistant]. It may be better to describe these tunings with numbers in order for them to be translated to other keys [135 or 1355, etc]. Bing Futch teaches some interesting things in his book 'Blues method for MD 101' that uses 'boxes'.

I've used a variation on Janita's tuning: D-A-d-A#.   Also D-A-d-c#.  I can mute the near string with the heel of my hand.  That means I'm playing D-A-d (that's easy and I can strum!) with an extra string I can play only when I need it.

Another chromatic tuning is D-F#-A.  See this site: http://www.mountaindulcimer-1-3-5.com/

With 4 equidistant strings, try D-F#-A-D. 

You can substitute a "power chord" when you can't get a major or minor chord.  A power chord contains only the root and 5th notes.  It's the 3rd note that determines whether the chord is major or minor.  When you play an open DAd you're playing a D power chord, which can substitute for either D-major or D-minor as needed.  Not the same sound, but it works in a pinch, especially when playing with other instruments that will supply the missing 3rd.

When dulcimer tab shows a barred 1st fret as an "E-minor" chord, it's actually an E power chord.  It's the kind of workaround that dulcimer players use all the time, but a guitarist won't know about. 

I think it's great that you've found a guitar teacher who's willing to explore dulcimer with you.  However, he may need an occasional gentle reminder that a dulcimer is not a guitar.  The typical lesson plan for beginning guitarists is not well suited for dulcimer. The dulcimer is NOT designed to play in multiple keys from the same tuning.  Nor to play chromatic runs.  That doesn't mean it's impossible -- but it ain't easy and it's very uncommon for anyone to attempt it.  The addition of a 1-1/2 fret doesn't really change things -- it makes a few more chords and scales possible, but it does not mean you can take the normal scale boxes and movable chords that you'd learn on guitar and translate them directly to the dulcimer.

If you're hunting for a Gb, G and G# in the same song, then you're playing a song -- or an arrangement -- that might be easy on guitar, but on dulcimer it's crazy advanced.  If your heart's set on that style of music, your life will be much easier with a full-chromatic fretboard.

Just a quick note on Janita Baker's use of the DAA#d tuning: you can't strum. She fingerpicks, and that A# string allows her to get all the chromatic notes, which is why she can play jazz and ragtime and stuff. But if you follow her tablature and try to strum across all the strings, it will sound terrible.  The tuning only works for her style of fingerpicking.

You can try 4 equidistant tuning of DAA#d. Janita Baker uses it in several of her books. She also uses several others, which I don't remember, for other keys.