If there is a civil rights movement today, it is mostly for gay people. There are clearly efforts to improve the status of African Americans, but it is not clear that they are a movement anymore and it is not clear that they are for civil rights.
African Americans today have civil rights. In the eyes of the law, they are equal to Americans of other races. There are, of course, problems that face the African American community. However, these are more of economic problems and less (with the possible exception of voting rights issues) about civil rights. Moreover, the efforts that are being made to improve the status of African Americans are mainly being done politically, through the electoral and legislative processes.
To the extent that there is a civil rights movement, it is centered more around issues of gay rights. Gays are the one major group in the US today whose civil rights (right to marry, right to adopt children, etc) are in some doubt. Even this movement does much of its work in conventional ways and may not, therefore, truly deserve to be called a movement. However, it is as close as we come in modern times to a civil rights movement.