Office : When multiple directors want you on their team and pick sides in a company that is going through reorganization, what t

So our firm is going through a reorganization that started about 1 year ago, it has hit its peak this year with quite a bit turnover. And for some reason, the skill sets I have had gotten attention in the company and now multiple directors and senior directors want me to be on their team or take their sides in meetings. I only report to one director but this manager has also had a few people reporting to him being reassigned to other directors in this past year so things are fairly unstable. I am not exactly sure how to handle dilemma situations like this as I don't want to pick sides. It is also hard to align my personal development directly to the person I report to now since we are in a fairly unstable stage right now. So anyone that my team has a conflict of interests with now may be the new team I get to be assigned to in the near future. So I don't want to really get on anyone's bad sides and end up on a bad project. How do you guys handle situation like this when you went through reorganization, especially in highly political environment like IB? Note that I don't have a choice to move to a new opportunity in current situation unfortunately. What would be a diplomatic way to managing up and managing multiple managers while getting things done?