He practiced law and became Pine Bluff city attorney, serving from 1968-1970. Dickey later served as a special justice of the Arkansas Supreme Court. He has been a business owner and also a lecturer at the University of Arkansas at Pine Bluff.
Dickey was elected as a Republican to the 103rd Congress and to the three succeeding Congresses, serving from January 3, 1993, through January 3, 2001. His run for re-election in 2000 was unsuccessful, as was another campaign in 2002.