The trick is to acknowledge these emotions in ourselves and dismiss them.
There are a number of ways that you can do this, and what works for you may vary from emotion to emotion and moment to moment.
For some, talking to their fears is a good way to open dialogue within themselves to figure out what is really bothering them. Sometimes, just acknowledging the fear is a good way to reduce its power.
Other people have great success with visualizing the emotion or situation that is bothering them, and then surrounding it with love, allowing it to transform into something positive.
If you do this and it helps, but the emotion is still present, do it again.
The key in all of this is to be judgment-free and aware that the initial emotion or issue that arises may not be the real issue at hand, but a symptom of something deeper that is ready to be healed.