The first banjos were made by slaves based on similar instruments from Africa.

The call and response used in African-American churches is believed to originate in call and response patterns found in West African tribal governance and religious practice.

The Br'er Rabbit stories are based on animal trickster stories of African origin.

You may also be interested in Geechee and Gullah cultures of the Georgia and South Carolina coastal islands, which speak creole languages incorporating many loanwords from African languages.

Some of these words have spread to wider usage the deep south, such as:

• the Bantu word okra
• gumbo, the Umbundu word for okra
• cooter - turtle, from Mandingo kuta
• goober - peanut, from Kongo nguba
• yam - sweet potato, from the true yam cultivated by the Igbo people

Much of modern music in America, and I'm speaking in broad strokes here, has roots in jazz or blues. These genres have roots in African slave songs; call and respond leads to gospel, blues, RnB, rock and roll, hip hop, and so on. The first episode of Ken Burns' Jazz has some good (if ken burnsy) info on the origins of black music in America, the roots of which have a direct connection to African music brought over by slaves, most famously centered in New Orleans' Congo Square.

A lot of modern black culture can be seen as a blended culture of circumstance: that is to day there are language and communal aspects of African American culture which exist and persist as a mix between the cultures retained from Africa by imported slaves during the early centuries, and the "American" culture they were forced into. What we see in the black church, for example, is a unique cultural interpretation of Christianity, which is by all other means a "white American" institution--the particular aspects of Christianity the black church stresses is a reflection of the struggles faced by the early (and therefore modern) black community, though again, this has as much to do with the specific struggles faced by black community once they got to America as it does the culture they practiced before arriving on the American continent.

I wish I had some specific resources in mind to point to where in depth analysis of this good question can be offered. There are many books on the cultural impact black music, politics, and community has had on non-black American culture. But as other posters in this thread have said, it is a very broad question with broader answers, given the mercurial nature of culture over time (as well in the complexities of trying to put into a rigidly defined box something as vast as "African culture"

Of course there are traces of African culture present in the US. For example, Louisiana Voodoo is a religious tradition that incorporated aspects of both West African Dahomeyan Vodun and European Catholicism. Obviously it's been incredibly commercialized and ridiculed throughout the 20th century, but there are some modern practitioners as well, and obviously stereotypes about it have entered American popular culture (ex. voodoo dolls)— from a modern voodoo website:

The main focus of Louisiana Voodoo today is to serve others and influence the outcome of life events through the connection with nature, spirits, and ancestors. True rituals are held "behind closed doors" as a showy ritual would be considered disrespectful to the spirits. Voodoo methods include readings, spiritual baths, specially devised diets, prayer, and personal ceremony. Voodoo is often used to cure anxiety, addictions, depression, loneliness, and other ailments. It seeks to help the hungry, the poor, and the sick as Marie Laveau once did.

Is there even such a thing as "African culture"? Seems to me as if there is simply way too much cultural variety in Africa to speak of a single "African culture".