The original question was how to play grace notes.   I am not sure it is helpful to bring in accidentals, since grace notes may or may not be accidentals. 

As Ken notes, grace notes are often ornamental in nature and not an essential part of the melody. As Robin explains they are usually written much smaller than regular notes in standard musical notation.

Grace notes always appear just before a note and they receive no counted value. The note just after the grace note sounds on beat, so the grace note actually takes away some of the duration of the note preceding it.

On a stringed instrument, a grace note would be plucked with the right hand, but the left hand would employ a slide, a hammer-on, or a pull-off to play the main note after the grace note. I do not think many of us could actually pick both notes fast or smoothly enough. But the important point, again, is that the note following the grace note falls exactly on the beat, so the grace note precedes it without itself receiving any counted value.

Below is the first line of my arrangement of the old Quaker hymn Beech Spring.  Notice the three grace notes.  In each case, I suggest playing it as a hammer-on.  If this were arranged for noter/drone play in DAA, I would suggest a slide in the first case and a hammer-on in the second and third since the grace note would be the open string.  And note that none of these grace notes are accidentals.

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Dusty T., Northern California
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updated by @dusty: 05/11/16 05:37:33PM

Now I may have this wrong so don't quote me!  But I think of grace notes as the quick augmentation notes that you hear around the melody in many types of music.  They are not part of the melody but usually just a passing flash of an adjacent note.  They tend to be in the key of the piece being played.  This tune I recorded this week uses a lot of grace notes on the downward melody runs:

Now 'accidentals' I think of as notes in a tune that are not 'in key'.  When written in notation they are the notes that have a sharp, flat or natural sign next to them on the staff to show that particular note in different from the key signature.  There are a few tunes I play that have accidentals.  Here is West Virginia Hills.  The tune is in the key of D and the accidental is an F natural.  The tune goes 'Oh those West Vir ginia hills' and the accidental falls on the Vir...

http://k003.kiwi6.com/hotlink/jxhmqvq1bo/West_Virginia_Hills_-_Jeffreys_-_15_Jun_14.mp3

 As I play with a noter there are certain techniques I use for both grace notes and accidentals.  I've outlined then in this video:

 

updated by @robin-clark: 03/03/16 06:55:34PM

Thanks Ken, I'd not heard the term "grace" before. As for accidental notes, simple key the string with the noter touching the fret board to play the half note. Robin Clark demonstrates this technique in his advanced noter video.  

Someone was recently asking about how to play grace notes.  That's one of those "depends" questions.  "Grace Notes" encompass a wide variety of musical ornamentation types. So, it depends on what kind of grace or ornamentation you're talking about.    In SMN grace notes are written in smaller notation to indicate that in many cases grace notes are 'fill ins' between notes used to fill up pauses or rests or sustained notes.

The most common dulcimer grace notes are those we use to cover the fact that the tune we're playing contains one or more 'accidentals' that aren't found on our fretboard in the tuning we're using.

As a traditional dulcimer player, I get the occasional 'accidental' when a Tab shows that I need to play a 6+ fret and my dulcimer simply doesn't have that fret. 

So how do you play a note that isn't there?  One way is to play the two notes above and below the missing note -- in the time of the missing note.  Say the accidental is a quarter note on the 6+ fret (and I don't have one).  In place of the missing note I would play the 6th fret and 7th fret as eighth notes.  Need a Bb and don't have it?  Play a B and a B# in the same time as the Bb is supposed to be.  If the tune is a familiar one, the listener will even hear the note that isn't there, because their brain expects to hear it.

We can also 'cover' a missing note using a grace note slide between the notes that are there, with the slide occupying the beat of the missing note.  A sort of syncopated hammer-on or hammer-off cane also be a grace note for something that isn't there.

updated by @ken-hulme: 10/27/19 12:02:25PM