For me, many Asian-imported anime/manga/fantasy stories contain names, storylines, body language, and visual styles are are just plain silly. By the same token, I know it's not necessarily any sillier than the Gnarley Forest, the Duchy of Geoff, or the House Orien Express, but the tropes just keep me from enjoying them.
The storylines, however, when boiled down to their essentials, are quite well done and adult in content. Consider a boy with a cursed destiny, shunned by many, who desires to become the heroic guardian of his people; one who takes that curse, and transforms it into something Good, not evil, and along his journey discovers what it takes to really be a hero, in real life rather than in stories. From what I can gather, that's the storyline of Naruto (from a friend of mine), but the actual on-screen look of the characters, the way the characters act, their power effects, and the unrealistic aesthetic, just puts me off. I had to have someone boil it down to appreciate the story, and I still can't bring myself to watch the darn thing more than a few minutes at a time.
About the only anime I liked was Cowboy Bebop, but mainly because it RARELY showed the kinds of elements that other Asian anime shows. It was grittier in tone (thanks to the gumshoe film-noir emulations), and I "got" the storyline. And not one person showed up with a 7-foot sword.

This coming from a guy who watched Ultraman and Space Giants as a kid - I just dislike the aesthetic, and when many of its elements show up in other things I enjoy, like D&D and Star Wars, it tunes me out immediately to even the good parts. Heck, to this day I still can't watch those Tartakovsky Clone Wars cartoons, because the style reminds me too much of it, for some inexplicable reason.