Basically, you can pronounce written Dutch three ways, (1) as if it
were German, (2) as if it were English, and (3) as if it were Dutch.
The last, while correct, is difficult for speakers of English, and
cannot be easily explained, even by native Dutch speakers, in an
Dutch sounds "funny" rather than "foreign," almost as if it were
a barely comprehensible dialect of English, or maybe German. In
particular, the Dutch letter G (it occurs in draaiorgel) is softer
than in English, with the tongue not quite closed against the soft
palate. Thus the German who transforms it to the German "Drehorgel"
and the American who says "dry-orghel" are both close, but both wrong.
I cannot give the genuine Dutch pronunciation here, because I do
not speak the language. (However, I sometimes listen, in immense
frustration, to Radio Nederland broadcasts on my short-wave radio.)
Perhaps draaiorgel is doomed to have a bad but commonly accepted
English pronunciation, just like dachshund (it really should be
dahks-hoont, not dash-hound), Des Moins and Des Plaines.