Most senior managers will have had to run some sort of enquiry or investigation at some point in their career. The best way to run almost any investigation is, in my experience, as transparently as possible. There are, of course, some fairly obvious exceptions - HR issues or those involving national security (I've been involved in the first of these, but not, thankfully, the second). Other than these sort of investigations, the investigation should be held as publicly as possible or appropriate.
What has brought this to mind is the UK Government's decision to hold an enquiry into the background to the Iraq war in private. Short of reasons of national security - and I can't see that there would be these sorts of reasons - I do not understand this decision. No matter how intensive, independent and intrusive this investigation is, because it is not held in the open, there will be concerns that something has been held back.
It is the same with enquiries, investigations or projects within an organisation. Tell everybody you can what is going on. Tell everybody you can who is part of the investigation. Tell everybody you can what your findings are. That way the chances are high that the findings will be accepted (assuming that you have done a decent job - and since it is so "public" then you are more likely to have done so) - even by people who do not agree with them.