Thank you, Lois!  I'll definitely check out that discussion.

@floralin, my sympathy when you compare the 3 stringed instruments where a chord's position is fixed.  At first this (coming from a guitar background) and the variety of tunings so that a particular fret may change notes (coming from a piano background) left me, too, as you said, "but when I try to memorize dulcimer chords my brain says, 'oh no, enough is enough'."

Dusty's given you the right advice.  I would also suggest you look at the discussion on "More DAd than DAA Instruction available."  You probably thought it didn't fit your needs and skipped it, but there are some points in it and in Strumelia's blog that may prove worthwhile for your exposure as you begin your dulcimer journey.

Yes, it definitely helps to know the names of the chords in a class where the teacher is discussing chord progressions.  ; -) 

You don't have to know what you're playing to play very well. But it makes it hard to communicate with others about music. If you want to ask questions or answer other people's questions, it helps to actually know the names of the notes and chords you're playing.

It is worth paying attention to the point your teacher was trying to make even if you don't always follow his or her advice.  There is nothing wrong with playing a 3-1-0 G chord with your thumb on the bass string, and it might work great if your next chord position is a 4-0-1 or 4-2-1 A.  However, if your next chord position is an A chord played as either 1-0-4 or 1-2-4, when you will need your thumb on the melody, then your whole hand has to change not only its position but its angle of attack.  However, if you play 3-1-0 with your index finger on the bass, then your thumb is almost on that 4th fret on the melody string anyway, so you are already in position for that 1-0-4 chord.

My advice is to be as flexible as possible and not get locked into a single approach.  Your thumb on the bass may be fine some of the time and also prove a barrier to smooth playing in other contexts. We all know that cliche to "go with the flow," and I would suggest that you want to adapt your playing to best capture the flow of the music.

Thank you all so much for your responses.  It was a teacher (I don't  remember who) who told me not to use my thumb, I think because it would take too long to reposition to the next chord.  I'm talking mostly about the 3-1-0  and 5-3-3 chords, in DAd tuning.  Or I think that's what  I mean, because...

Another little problem I'm having is that I know guitar chords, know banjo chords, and can fake mandolin chords (have trouble with the barre), but when I try to memorize dulcimer chords my brain says, "oh no, enough is enough".  So I'm having to go by how the chord feels in the piece or sit there and say, "let's see, d-e-f#-g, and a-b, and d, so this must be a g chord".  This is slow.  But I can play a piece all the way through without remembering the names of the chords, or choose to use a chord without knowing its name.

I've had classes with both Stephen Seifert and Aaron O'Rourke.  They are wonderful!

@Floralin, I personally never use my thumb on the bass string, but I know some superb players who do so a lot.  I also have one student who does it even on some of my own arrangements.  The short answer is if it works for you, then it works for you.

One big issue when it comes to left-hand fingering is how the dulcimer is positioned on your lap.  People who use their thumb a lot will want the dulcimer to be angled so that the lower bout is tight against their right hip and the head is angled out over the left knee (assuming you are a righty). Watch some of Guy-Babusek's videos here to see a good demonstration of that kind of positioning.  That angle makes it much more comfortable to use your thumb (but much harder to use your pinky).  People who don't use their thumb as much and rely more on their pinky move the head of the dulcimer in more so that the dulcimer is closer to being perpendicular to their legs.  That positioning facilitates the use of the pinky.  Check out Aaron O'Rourke's videos for an example of that, and notice that he also lifts the bottom of the dulcimer off his lap and angles it away from his body, something else that would make it harder to use your thumb on the bass string (and maybe the reason I never do it).

Personally, I use both my thumb and my pinky, and I shift the dulcimer on my lap depending on the fingering for any given song.

I am not sure which G chord you are talking about.  In DAd:

I play the 3-1-0 G chord with my index on the bass string and either my ring finger or my pinky on the middle string. 

For the 0-1-3 chord I use my ring or middle finger on the middle and my thumb on the melody. 

For the 3-3-5 chord, I use either index, middle, thumb or middle, ring, thumb.

For the 5-3-3, I use index, ring, and pinky. 

For the 5-6-7 chord I use either ring, middle, thumb or middle, index, thumb.

For the 7-6-5 chord I use either index, middle, pinky or index, middle, ring.

I'm sure there are other ways to get these chords (in fact, I know both Stephen Seifert and Aaron O'Rourke barre two of the strings when they play the 3-3-5 and 5-3-3 chords) but those approaches work for me, and in cases when there are more than one possibility, the context of the notes before and after will dictate which I use.

My advice is to do what works, but be aware that it might not work all the time, so be prepared to try other possibilities as well.

You'll get lots of opinions on this, probably best answered by someone who teaches in chord/melody style.  :)

Of course a lot depends on 'who' told you this (another beginner student? the teacher?..), and what the reason was for their advice. There might be a good reasons to follow or not follow their advice based on the situation. But a good teacher will usually have a good reason for their advise. Maybe using the thumb there create an awkward hand/wrist angle.

I'm assuming you're in DAd tuning? There is more than one way to finger a G chord. Which G chord  are you referring to?
A lot also depends on the sequence of chords or notes going on in a particular tune as well. Sometimes finger positions will be changed from the norm in order to make a specific musical phrase or sequence flow better and not sound choppy.

Various factors at play on this. Hopefully an actual chord style teacher will jump in to help here.

I'm not a Chord-Melody player.  But the first rule of playing dulcimer (having done it for 40+ years) is this: 
 There Is NO right way, or wrong way to play the dulcimer -- there is only The Way that works for you. 

Others may not like or approve of how you do play, but that's their problem.  Let no one dictate to you. Just because someone is a teacher does not make them right -- for you. 

I play other instruments but am fairly new to the dulcimer.  I was told in a dulcimer class that using the thumb on the low 'd' string - for example, when making a G chord - is a no-no.  But sometimes it just seems more convenient and quicker.  Opinions?