I found an interesting entry on a blog I follow, about a "new" method of introducing calculus, by Donald Knuth. Fundamentally, it involves introducing the big-oh notation to define a "strong derivative", and recovering most results including the Fundamental Theorem of Calculus (see the comments section). The resulting math feels light - like you were doing a back of the envelope calculation.
It seems interesting, and although I am not sure whether that is the magic potion that will enable all my students to master the idea of applying calculus in physical problems.
Personally, I never had a problem with the traditional approach starting with the definition of a limit.