I am a very big fan of cilantro, I even grow my own I use so much of it. Cilantro is a strong flavor and people either love it or hate it. I've never seen anyone shrug and say they can "take it or leave it" or "leaf" it, pardon my really bad pun.
This martini reminds me a little of a gimlet, the lime is a co-star but, of course, secondary to the cilantro. The muddled cilantro infuses it's aroma and flavor into the mix when blended with the lime juice and sugar in the shaker.
I don't strain my Cilantro Martini at all - I love the little bits of cilantro that float around in the martini visually as well as enjoying those little bits bursting flavor on my palate.
I truly love this martini, I find it a nice change from all the sweet and creamy martinis I like. I even do it in a version with ginger and with dill, but this one remains my favorite.