What are the movements of the 20th century that have most impacted African American religious culture?
The most important movement that impacted African American religious culture in the 20th century was the Civil Rights Movement. Without even thinking, one can name a number of highly religious African Americans, most notably Martin Luther King, Jr., who became role models for the culture. Not only did these religious leaders influence African American culture, but American culture as a whole!
Why is this so? Because in Christianity, and especially in the Protestant sector of Christianity (where most African American Religious culture resides), the black Protestant churches even in the time of slavery and up to the present always stress two things: equality in God's eyes and peace in God's world.
I have decided to stick to love. ... Hate is too great a burden to bear. - Martin Luther King, Jr.
In my opinion, this impact of the Civil Rights Movement on African American religion continues even today with leaders such as the Reverend Al Sharpton and the Reverend Jesse Jackson. Because of the importance of both equality and peace, one can see why the American Civil Rights Movement is THE most important movement in regards to your question.
However, let's not restrict African American religious culture to Christianity, as there is a large percentage of Islamic African Americans as well! Although there isn't a specific movement specific to the 20th century that is pertinent, there are a few things we should talk about. First, that Islam was accepted through much of Africa during the slave trade due to the instrument of PEACE in its spread. No crusades where Islam is concerned on that continent of Africa! But in regards to the 1900s, probably the closest thing to a "movement" was the influence of Wallace Fard and his creation of the Black Nationalist Nation of Islam. This happened during the Great Depression of the 1930s and inspired many African Americans to become Muslim. If we can use Malcolm X as an example, even though he ended up converting to traditional Islam, he BEGAN in the Black Nationalist Nation of Islam.
In conclusion, even though Protestantism and Islam include the main movements of the 20th century in regards to African American religion, other religions are represented as well. The influence of Voodoo, especially in regards to the influx of Haitians in the middle of the 20th century, is significant, ... especially in the state of Louisiana. Therefore, although not a "movement," it is certainly worth mentioning. Further, anyone who likes any kind of Gospel music needs to thank the origins tracing all the way back to Africa, as this kind of music can be said to have enriched all American religious culture.
The movement that impacted African American religion most was perhaps by Marcus Garvey, Jamaican political activist. He founded the UNIA, Universal Negro Improvement Association. This association was Christian, and they believed in a 'black' Christianity, with a 'black' God and the 'black' Madonna. This contributed to later movements that occurred in different religious African American movements.