I know that this question might seem silly, but you always hear about civil rights, and I don't recall ever hearing about rights in court. I'm sure it was different in each state, but did blacks lack access to the court system in the South?
Many black Americans, often in conjunction with organisations such as the NAACP did use the court system to try and seek change or better civil rights. One notable example was Homer Adolph Plessy v. The State of Louisiana in 1892, which challenged the Louisiana Seperate Car Act law of 1890 which had demanded seperate but equal accomodation on railroads for the different races. This was taken to the Louisiana Supreme Court, which upheld segregation, and then the US Supreme Court as Plessy v Ferguson in 1896. The Supreme Court supported legal segregation in a 7-1 decision.
Other examples pre-MLK of blacks using the court system to incite change include Williams v Mississippi of 1898 (disenfranchisement), Cumming v Richmond County Board of Education 1899 (segregation in schooling), and the famous 1954-55 Brown v Board of Education of Topeka, Kansas, which overturned Plessy v Ferguson's "seperate but equal" notion and declared segregated education unconstitutional.
So yes, there were numerous instances of African-Americans using the court system to push for civil rights. Unfortunately, most were unsuccessful, particularly on a state-level, facing racist Judges and all-white juries. Furthermore, without the support of organisations with funding, individual black Americans often couldn't afford to take their issues to court anyway.