Earlier today, the Braves announced that they'll be leaving Turner Field in favor of a new stadium in Cobb County, north of Atlanta. The move caught people off guard in part because of Turner Field's age—just 16 years old!—but also because it represents an enormous shift toward suburban baseball on a scale not seen in decades.
The chart above shows 19 MLB stadium moves from 1960 to present—including the impending Braves move—with the lines measuring how far, as the crow flies, the team shifted toward or away from the city center (defining the "city center" can be tricky; I deferred to Google Maps.) As you can see, the recent trend has been to move parks into the inner city; Oriole Park at Camden Yards in Baltimore is an oft-touted examples. Of the 18 stadium moves shown above, just five took teams farther from their urban core, and, of those five, four were shifts of less than a third of a mile. Kauffman Stadium moved the Royals 4.7 miles farther away from downtown Kansas City—the Cobb County stadium will move the Braves 9.7 miles.
To set some minimum for what counts as a "significant" move, I didn't include new fields built within a fifth of a mile of the previous field.* Likewise, I didn't include moves from temporary stadiums to permanent ones (for example, the Astros played in Colt Stadium while the Astrodome was under construction), although I did include the Kansas City move. While the earlier park—Municipal Stadium—was just a temporary site for the Royals from 1969-1972, it was the Athletics' park from 1955-1967, so I felt that this still represented a significant change for Kansas City baseball. If you take it out, the Braves' decision just sticks out that much more.
*These are: U.S. Cellular Field, Miller Park, the Great American Ballpark, Citizens Bank Park, Busch Stadium III, Rangers Park at Arlington, Yankee Stadium II, and Citi Field.