I agree with the other posts that doing a big blitz up front will likely put people off.
I manage about 40 people in a couple of different areas. What I have done is to just start using the language and techniques--gradually make it part of the way you do business with people. Here are some suggestions:
"So what is the next action we need to take on that?"
"What are the projects you are currently working on?"
"What are we trying to accomplish with this?'
I recently was handed a work group to manage after two other leaders of this group left. We had a significant inspection coming down from on high and needed to get organized fast.
I stood up at the white board in the conference room and asked the team, "I need to get up to speed on things here. What projects are we working on, or need to get worked on?" (They did a mind sweep and didn't even know they were doing it).
Once I had a list of projects, I went back and clarified each of them with an outcome and then asked for next actions and who was going to be responsible for it. We prioritized the projects on the ones that needed to get done now and the ones that could wait until after this big inspection.
It was simple, fast, no nonsense. A group that was sort of drifiting without leadership suddenly got energized. All I did was basic GTD stuff and I never said anything about GTD.