If you've never searched for your own name on Facebook, I recommend it. I found a comment that a former student wrote about me on a current student's page. She was complimentary, saying she'd like to take more classes with me, because I'm "a great storyteller."
I never consciously decided to make story-telling one of my primary methods of teaching. That comes out of my own interest in stories. Before college, I wanted to be a history major. Most of my high school classmates viewed history as a boring collection of dates and facts, but to me it was a collection of stories. People's lives are endlessly fascinating, and real life is more unique, idiosyncratic, and fun than fiction, if only, as Twain said, because fiction has to make sense.
Embedding the perfect story into your lecture does three things for your students: 1) it wakes them up, by interrupting the normal rhythm of the lecture; 2) it engages them with the material, since story-telling is more intimate than "just the facts" lectures; and 3) it helps them remember concepts through applied examples. Add a story to your lecture, and I guarantee you will find that story repeated in half the essays on the next exam.
Usually the stories I tell are true (or true enough), but I've experimented with students telling fictional stories as a pedagogical technique. For the final project in my Latin American archaeology class, students have the option of writing a standard term paper or creating a short, fictionalized account of a prehistoric context, one that brings the archaeological record to life in a way that no research paper ever could.