Eoin O'Neill, bouzouki player from Clare (though originally from Dublin, he's now fully integrated in the Clare session scene) and present on many productions from artists of this region (Mary Custy, Terry Bingham, Kevin Griffin, etc.) is a very prominent user of the bouzouki in ADAD tuning. Effectively, this tuning could be called "open", even though there is no full chord here (a chord that deserves it's name exists of at least 3 notes): it could be considered as an open D tuning, but without the 3rd step, the F#. Since most of the Irish trad. music is in the key of D this should be a most convenient tuning for it. And it is.
Basically, simplicity rules. Since you already have a D chord without fretting any string, you could strum and drone behind any tune in D, and not sound out of place. Then, by simply adding any note in the D scale, you could colour this D chord to sound like something that will effectively do the job of any chord you'd want for the scale of D. This is what Eoin does most of the time: he knows his D scale in and out, up and down, and he just colours his droning D chord. He occasionally throws in a single note run, either following or countering the melody, but if you know your scales, that's a piece of cake, of course.
Every now and then he will then go for some more complex "chord" forms, where he even frets 2, or in one extreme (:-)) case, 3 strings. Here's a short (but comprehensive) list of the chord forms I saw him use during the session I observed (and enjoyed):
A D A D 5 4 5 0 Dmajor 5 4 0 0 Dmajor 0 4 5 0 Dmajor/A bass 5 0 5 0 All D's 0 2 4 0 Amajor 4 2 0 0 Amajor/C# bass 2 4 0 0 Bminor7 0 2 0 2 A, no 3rd 5 3 5 0 Dminor 5 3 0 0 Dminor 0 5 2 0 Gmajor 0 7 4 0 Amajor 0 2 3 0 AminorUp to now we discussed only the key of D. But what to do for other keys ? Eoin uses the easy way out here: he just slips the capo (which is standby just behind the nut - he uses one of these rubber band contraptions) up to the appropriate fret, and continues to use the same approach as for D. The only drawback is that you can't really work in the key of C - it would mean you'd have to go too far up the neck. But all the common keys can be covered: G = capo 5th, A = capo 7th, Em = capo 2nd, etc.
Eoin's right hand works mostly from the wrist, and he rarely strums across all 4 pairs of strings. When he does, he makes somewhat longer strokes, using the forearm also. But most of the times he just walks up and down the strings, sort of pattern picking in a style not unlike what Alec Finn does on his 6-string bouzouki. And indeed, he uses mostly the top 3 pairs, not all 4, so that he has the lowest D string as his starting point. It's hard to explain this exactly in text - you'd have to see and hear him to get the point.