On the other hand, I do have a friend who, while he was Dean, had his email on three day delay. The theory: only those problems which were still problems after 72 hours were worth addressing at a super-departmental level.
I'm a professor. I can't help but think that this is a terrible idea.
As Dean, it is your job to enable the faculty members and department chairs who report to you. To figure out what obstacles they face, and remove them when possible. To figure out who is doing good work -- especially of the sort that goes unnoticed. To sort through requests for money and other resources, and to figure out who will put these resources to good use.
In other words, the same as any manager at any organization.
The medium is part of the message. Avoid sending the wrong message.
Note that immediacy and importance are orthogonal.