On TV every day there are anime shows Americans watch. Also, Americans (I don't know about other Western countries, I'm American...) often say "Sayonara" (incorrectly, but we attempt, nonetheless), which is the Japanese word for goodbye. Asian languages like Korean and Chinese are becoming more popular in American schools. Hmm...What else?
Ahh, yes...If you were to go to an American book store, there's always a section for anime. Comic books that aren't traditionally American (Superman, Batman, aren't they American...?) have become more popular...Inuyasha, Bleach, or Death Note, for instance, come on TV and are comics...Ranma 1/2, Tokyo Mew Mew...You name the anime, you can probably find it somewhere in an American shop.
Finally, we have restaraunts all around the country that are Chinese, Vietnamese, Thai, Korean, Japanese, Indian... Foods from Asia can be found in grocery stores, too. Hope that helps ^^