They enjoy the grand, epic style of cartoons like Yu-Gi-Oh which pit the forces of good against evil and which have a continuous story-line that builds throughout the series. Japanese series have a knack for that type of story-building and mythology-making that most Western cartoons don't develop nearly as well. Most Western-style cartoons tend to have more stand-alone episodes and tend to keep only a thin thread of mythology running through the series.
Currently, I am being pestered by my children to read the Warriors books, a series of books about warring cat clans. It sounds a little weird...but they love it. They read each book and discuss it afterwards. They run around outside pretending to be from warring cat clans. They talk about the special terminology used in the books. They whisper about the parts I haven't read yet, so as not to spoil the surprise. They talk about the twist and turns in the books.
In short....they love these books.
These books do the same thing that a good anime series does; they develop a consistent, ongoing, mythical story-line while incorporating adventurous battles and intrigue.
So...if you have a child in the upper elementary grades, or lower if they're advanced readers, give the books a try.