Perhaps the biggest thing a manager can do is show they are a part of the team rather than apart from the team. As a leader, you are saddled with a truckload of responsibilities, they take up your time and stress you out. They are also important, after all, they're your responsibilities. When you become consumed by them however, you appear to be detached from your team. You give off the, "I can't help you today" vibe. Build time into your day, every day, to spend time helping your team with their tasks. That might not mean doing the tasks for them, it might mean entertaining customers while staff catches up on the volume, answering a few phone calls on the main line, making copies of a log or fax cover sheet when you notice the supply is low, handling a task that you already delegated to them but they haven't yet completed, or a host of other things. When you show your staff that you are willing to help pick up the slack in the most basic of team tasks you show that you hold yourself accountable to the completion of those tasks the same way you hold them accountable.
Another big opportunity for leaders is to show your human side. No, you don't have to cross the professional-personal barrier. Simply talking about a project you are working on, or some light venting about a deadline you have to meet for a report shows them that you're part of the system too (I'm not meaning only corporations, but all systems where a team is working towards a goal). I feel a lot of leaders fear they will appear weak in front of employees when they do this, but sharing your feelings helps employees identify the fact that you have a job to do too. When they make that realization, they'll cut you a little more slack when you spend time doing it.
I'm not going to find the right way to word this, but I'm open to any discussion that leads to better understanding of viewpoints. The most important thing that a staff member can do is to accept that management is part of the system. Again, not related only to corporations, but to the process of completing the tasks. Management may progressively have more power and authority the higher up the hierarchy you get, but they relationship can still be flat. This begins by holding managers accountable to their responsibilities and involvement with the team. I am not condoning documented coaching of your manager, but rather keeping your expectations in focus in a professional manner.
When teams enact more of these attitudes within their daily activities they'll begin to work more closely together. This can open up better communication which helps teams work towards their vision.