"Which "r" of "surprise" do think is silent?!"
I think to many Americans the first syllable of "serpent" has a noticiable "r" at the end whereas "surprise" doesn't, so that is what they mean by the missing r. To me they both have more-or-less the same first syllable, so I suspect I would sound like I am not saying the "r" in "serpent" to them as well.
As I said in another post on this thread the technical term is "rhotic" for the pronunciation which says the "r" in "serpent", "non-rhotic" where you don't. Wikipedia has an interesting article on it http://en.wikipedia.org/wiki/Rhotic_and_non-rhotic_accents. Are you a non-rhotic speaker too?