A native of Alexandria, Virginia, Dudley majored in chemistry at the University of Virginia. He turned to broadcasting in the late 1930s, starting out at a Charlottesville radio station. He moved up to calling Chicago Cubs and Chicago White Sox games from 1938-1941 before serving as a pilot in the U.S. Army Air Forces during World War II.
Dudley was the Indians' lead announcer from 1948 until his firing by the club in January 1968. In 1969, Dudley broadcast for the expansion Seattle Pilots; when the club moved to Milwaukee and became the Brewers the following year, he did not join them. Dudley broadcast for a number of minor league teams in the 1970s before retiring. As an announcer, Dudley was known for his friendly, homespun style and his signature catchphrases: "Hello, baseball fans everywhere" (to start a broadcast), "The string is out" (describing a full count on a hitter), "A swing and a miss!-he struck him out", "That ball is going...going...gone!" (to describe a home run) and "So long and lots of good luck, you hear?" (signing off at the game's end - "you hear" sounded more like "ya he-ah?" in Dudley's vocal inflection).
Dudley was also a popular advertising pitchman in Cleveland, remembered primarily for his radio and television commercials for the Aluminum Siding Corporation (Garfield 1-2323) and Kahn's Hot Dogs - "the wien-ah the world awaited," in Dudley's unique parlance.
In addition to baseball, Dudley also broadcast football at various times for the Ohio State University, the University of Washington, and the NFL's Cleveland Browns, Detroit Lions and Baltimore Colts.