Eric Thames, Rajai Davis, and Jayson Nix of the Blue Jays hit three consecutive triples in the fifth inning of last Wednesday's game against Cleveland.
According to this BRef post, the most recent games with three consecutive triples are:
June 8, 1961, Athletics, Top 3 (G1) - Leo Posada, Norm Siebern, Joe Pignatano
April 25, 1981, Cubs, Bottom 4 - Steve Henderson, Leon Durham, Jerry Morales
May 6, 1981, Expos, Bottom 9 - Mike Gates, Tim Raines, Tim Wallach
(In that Expos game, Andre Dawson followed the three triples with a double. In the 1961 game, Hank Bauer doubled before the three triples. On September 25, 2010, the Rockies had three triples and a home run in four straight batters: Payton 3B, Gonzalez 3B, Tulowitzki HR, Mora 3B.)
The record for consecutive triples in a game is four, set by the Red Sox against the Tigers at Fenway Park on May 6, 1934. In a 12-run fourth inning, Carl Reynolds, Moose Solters, Rick Ferrell, and Bucky Walters tripled. Boston's five triples that day (Reynolds had two) are the most any team has had at Fenway since 1919 (earlier than that, play-by-play data is incomplete).
The only other game (since 1919) in which the Red Sox hit more than three triples at home was on June 30, 1938, when they hit four. Including road games, Boston has hit four or more triples in a game ten times since 1919, though only twice since 1938.
On June 29, 2010, Minnesota's Denard Span hit three triples against the Tigers. It was the 29th time since 1919 that a player has had three triples in a game, though only the fifth time in the last 30 years. The MLB record is four triples, set by George Strief of the Philadelphia Athletics (American Association) on June 25, 1885 and Bill Joyce of the Washington Senators (National League) on May 18, 1897. In Strief's other 43 games that season, he tripled only once.