Chicago renamed South Park Way for King within four months after his death, and that's thought to be the first example. Austin (circa 1975) was also quite early. Most renamings occurred after the 1983 designation of a national holiday.

It's been reported (in today's Chicago Tribune) that calls to rename a longer street that traversed more of the city were resisted by Mayor Richard J. Daley because he feared vandalism of the signs. Today's rosy "After-School Special" views of Dr. King's work gloss over how intensely controversial he was at the time, especially his anti-poverty activism and opposition to the war in Vietnam. As Michael Eric Dyson puts it, most of America chooses to remember King as the “moral guardian of racial harmony” rather than as a radical challenger of the racial and economic order. When Chicago renamed the street, the pastor of Chicago's largest black church (Olivet Baptist) changed the building's address to a side street. In more recent decades, there has been resistance to name changes in white or business areas. Perhaps you've heard the joke that exit signs reading "MLK Blvd" serve to tell out-of-towners where they can buy crack.

The best-known article on the subject

Derek H. Alderman, “A Street Fit for a King: Naming Places and Commemoration in the American. South,” Professional Geographer 52:4 (2000): 672-84.

doesn't seem to be available online, but a later revised version is. Alderman found that census tracts containing streets named for King were on average 65% black, in cities that averaged 28% black populations.