Yes, they did. But they never faced the severity of discrimination that blacks did. And, more to the point, the discrimination against them ended. It hasn't ended against blacks.
Discrimination against Asians hasn't ended. Far from the truth.
In fact, one of the main issues Asian-Americans face is the fact that people
think that discriminatino against Asians has ended, that Asian-Americans, as the model minority, have proven themselves to be well-adjusted and assimilated. In truth, discrimination against Asian-Americans tends to be subtler, so it can be harder to spot - but it still hurts and is still pretty annoying at times.
Here's an excerpt from
an article concerning the issue (the author is a UC sociologist, for what it's worth):
Studies that take into account educational achievements find that Asian men earn less than their white male counterparts. Sociologists Chang Hwan Kim and Arthur Sakamoto found that if you compare white men to Asian men with similar characteristics, the white men often earn more. In other words, if an Asian American man and a white man both live in New York, both went to selective universities, and both studied engineering, we could expect that the Asian American man would earn, on average, 8 percent less than the white man.
[...]
Labour market discrimination against Asians is not unique to the US. A study conducted in Australia also uncovered labour market discrimination against Asians. Alison Booth and her colleagues conducted an audit study where they sent 4,000 fictitious job applications out for entry-level jobs, where they varied only the last name of the applicant - thereby signalling ethnicity.
The results were that the average call-back rate for Anglo-Saxons was 35 percent. Applications with an Italian-sounding name received responses 32 percent of the time - with only a small statistically significant difference. The differences were starker for the other groups: indigenous applicants obtained an interview 26 percent of the time, Chinese applicants 21 percent and Middle Easterners stood at 22 percent. According to these findings, Anglo-Saxons would have to submit three job applications to have a decent shot at getting a call-back whereas Chinese applicants should submit five.
And here's
a short article about a Cornell psychology study on the subtle forms of discrimination Asian-Americans face:
The researchers found that approximately 78 percent of the participants reported some form of racial microaggression within the two-week time frame. [...] The researchers also found that racial invalidations (e.g., being treated like a foreigner or overhearing racially biased sexual stereotypes) were more prevalent and harmful than racial microinsults (e.g., being told an offensive joke or comment concerning how Asians talk).
In fact, I'd argue that Asians might have it worse in the media. Asian males are always asexual, nerdy and geeky overachievers; Asian females are exotic beauties. Certainly, there are negative stereotypes that often pop up for other groups, but the difference is frankly it's hard for me to think of any depiction of Asians other than the negative stereotypes - I can still, for instance, think of more positive (or perhaps more "neutral") depictions of Africans. Racism towards Asians in the media doesn't generate controversy the way that it might for other groups. There was an SNL skit from a couple of years ago that dealt with this pretty well - it was during the height of the "Linsanity" surrounding Jeremy Lin, and the skit revolved around sports commentators making puns and other seemingly inoffensive but racist remarks towards Asians, but the moment one of them makes a similar remark concerning an African-American basketball player, he gets lampooned by his (hypocritical) colleages.
These media portrayals have had a definite effect on how people deal with us Asian-Americans - Asian-American males tend to have a difficult time hooking up with non-Asian females, while Asian-American females suffer the opposite, being idealized to the point of being creepy by non-Asian males (which, in turn, puts non-Asian males who genuinely love their Asian-American girlfriends/wives/etc. in a difficult position sometimes when their love is questioned, but that's another matter).
I've suffered this sort of subtler discrimination myself, but the fact that it was more subtle made it harder for me to grasp at first. For example a couple of years ago at the place I interned there was a girl (white or hispanic, I forgot), about my age, who also interned there. Whenever we talked about things, she would always make assumptions that I had no life, that I lacked social skills, etc., negative stereotypes associated with girls. In fact one time while we were talking with another intern she asked us something like "So what are your plans for the weekend? Doing anything interesting?' and immediately after she turned to me and said something like "Oh wait of course you won't be doing anything" and the other guy (hispanic, I think) glanced awkwardly at me as if he understood how hurtful her words were, but otherwise didn't say much. She also occasionally explained why she never wanted to date Asian guys, and her reasons were pretty much based off the media stereotypes, i.e. too focused on academics, too effeminate, not social enough, etc.
Asians have, and still are, ultimately perceived as an outsider. Even African-Americans, to some extent, are considered part of America - perhaps not perceived as a positive part of it, but a part of it nevertheless, at least more than Asians. We're always the people who can't pronounce English, who have a strange and mystical Oriental culture, whatever. The "Ching Chong Ling Long" incident from a couple of years back shows these perceptions are still alive and kicking.
I suppose this rant is somewhat off topic, and a bit of a rant, but I figured I'd get it off my chest. I want to emphasize that I am not saying Asians suffer equal or more discrimination than blacks or Hispanics or native Americans for instance (side note: I find it a bit amusing that native Americans are lumped together with Asians in questionnaires and surveys sometimes). I just want to say that Asian-Americans are far from doing well, as I mentioned in an earlier post while some studies have shown that Asian-Americans do well, the problem with these studies is that they don't consider Asian-Americans as a rather diverse group, and when you start breaking things down the situation gets much more complicated.
(I'd also think that Jewish-American probably still have their own issues to deal with, but I won't comment here since I don't think I am qualified to comment. (For what it's worth, though, I grew up in an area with a lot of Jews and Asians.))