A mere glance at the etymology of the word "democracy" (temporarily setting aside "direct" for now) reveals perhaps just how questionable it ought to be for anarchists.
"....from dēmos 'the people' + -kratia 'power, rule'."
"1570s, from Middle French démocratie (14c.), from Medieval Latin democratia (13c.), from Greek demokratia "popular government," from demos "common people," originally "district" (see demotic), + kratos "rule, strength"...."
So if the "common people" seek to "rule" collectively, then that too easily becomes a fixed tendency towards centralizing social power into a state or institution (a proto-state perhaps?). Never mind, for now, the usual ploys of collective decision making 'needs' and processes often ushered in as justifications. Individual need not "rule" themselves in an otherwise free society. I think anarchists would rather achieve self-determination, not self "rule".
The word "rule" also implies something "indirect" therefore DD seems conceptually incoherent. If it's so direct, then why bother to formalize any decisions at all?
Lastly, after reading Bob Blacks pamphlet http://sfbay-anarchists.org/wp-content/uploads/2013/01/Bob-Black-Debunking-Democracy.pdf
(which I definitely recommend), I was left with an overwhelming sense of just how ambiguous, abstract, and bewildering democracy is. Reading about the historical developments, the different types, and attempts at making democracy work, and how each attempt only ends up creating more problems (e.g. authoritarian manipulation, inequality) provides a real sense of context. To me that context is like a labyrinth of social relationships, out of which such supposedly 'noble' ideas are used by Statists to maneuver into positions of power.