What part of the body is the seat of emotions?

Symbolically, the heart but in reality, the medulla oblongata or the hindbrain.

Shown in research to perform a primary role in the processing and memory of emotional reactions, the AMYGDALAE are considered part of the limbic system.

https://en.wikipedia.org/wiki/Amygdala

The hypothalamus-medulla oblongata. Emotion and pain receives extensive attention in neuropsychological research. Research has indicated that the induction of positive emotions, such as happiness or mirth, results in significant changes in heart rate. Negative emotions are associated with alterations and anger and hostility are due to an increase in heart rate.

Overall it's the intestines, but the liver governs anger, the kidneys fear (the adrenals sit atop the kidneys) the lungs sorrow/depression. The stomach/spleen anxiety/worry, etc. But overall it's the intestines, ie. ("intestinal fortitude" a phrase coined by Dr. John W. Wilce professor of preventative med at Ohio State College of Medicine in 1955.) As crazy as this all may sound to some, it was common knowedge to the ancients, lost on modern Western medicine, but slowly being rediscovered. The reason that the medulla oblongata is thought of as "the seat of emotions" is that that is where the feelings or "symtoms" if you will, register. There are a few doctors on the cutting edge that are starting to understand this such as Dr. Mark Hyman.
See his videos on youtube -
http://www.youtube.com/watch?v=cd-jelcz6vg&eurl=http%3A%2F%2Fwww%2Eultrawellness%2Ecom%2Fblog%2Fmagnesium%2D0&feature=player_embedded