Alright so I'm seeing alot of extremes in the answers regarding this issue, either that Japanese people are the most exclusive, racist on the planet or that they're 100% totally and completely accepting no matter what.
First of all: racism, prejudice, and discrimination exist all over the world. Japan is no exception. Being African American, I'm sure by now you know that racism is widespread in your own country, wheather you have experienced any directly or not.
That being said, you must keep in mine that Japan is an island, and thus stayed relatively isolated and ethnically homogenous for quite some time. So you can imagine the shock of the native Japanese when, all of a sudden, there's people that look nothing like them roaming around. Of course they are going to be suspicious of ANY newcomer to their country.
Nowadays, especially the younger generations, are for the most part accepting or at least tolerant of foreigners. Older people may not be so fond of them. I say "mostly" and "may" because there are always exceptions to any generalized statement. Of course there will be people stereotyping you, gawking at you, even trying to ask you questions. Foreigners in Japan are perceived through a spectrum of different feelings and views. Again, it all depends.
To address what ♫Ami♪ said:
"It's offensive of you to ask such a question. Japanese people in general aren't racists."
"I've never met racist Japanese people :-O If I had, then, well, I hadn't...so, Japanese people aren't stereotypical at all."
"Trust me, Japanese people don't judge on races. Heck, whoever made that stereotype up is juscrazyy. I think THEY'RE the ones who are stereotypical and judgmental."
First of all it is not offensive to ask such a question. It is out of curiosity and concern that she asks it.
Secondly, I will say this about Japanese people: they do have a knack of following stereotypes, often to excess. As another poster said, they do often associate black people with gangsters or hip hop culture. When black people, and other foreigners appear on Japanese variety shows, even if they are speaking (near perfect) Japanese, will be given subtitles under the pretense that no Japanese person can understand what the heck they are saying. To add insult to injury, I've seen some variety shows that, as a joke, subtitle entirely in Katakana to show a slow, simplistic manner of speech in their subtitles.
Black people aren't alone though; the Japanese play up the stereotypes of many groups, racial or otherwise; look up Hardgay to get an idea of how they view American gay culture. x]
And of course, as I said earlier, the statement that "there's no such thing as a racist Japanese" is completely untrue. Of course there are racist Japanese people.
One problem in Japan that I find quite distressing is the open discrimination of some foreigners when it comes to some hot springs, bath houses, bars, and clubs. Some of these establishments will post signs outside the door denoting which customers are welcome and which ones are not. For example, a sign might read "No Filipinos" or "Japanese only". This is something that should not be happening in such a developed and advanced nation as Japan, but unfortunately, it does.
If you know a good deal about Japanese culture and customs, and if you try your best to speak Japanese, then you're that much closer to gaining their respect and acceptance. Japanese people are very polite, that is something they pride themselves on and I believe it to be very true.
However, they will always treat fellow Japanese and foreigners in two different ways, and if you wish to get a little less "tourist treatment" and a little more "authentic" treatment, then the Japanese knowledge will help. ;]
About "acting kawaii" - this all depends on your actual nature and personality! Across the globe, regardless of race, there are girls who have cute mannerisms, styles, ways of speaking, etc. A tell tale sign is if your friends frequently say "Oh my god, you're so cute, _________!!!"
If you are not normally cute, but rather sporty, punky, laid back, loud, boyish, or whatever - then for heaven's sake, don't start purring, squinting your eyes, doing peace signs left and right, raising your voice to a squeaky pitch, or actually saying "KAWAII DES~~~" to everything you see. Act normal. Embrace the culture but do not mock it.
Happy Travels!
P.s: I highly reccomend reading Gaijin Smash, a humourous blog written by an African American about his day to day life and experiences in Japan.
http://www.gaijinsmash.net