Beckham was referring to the flap that arose this winter after former closer Bobby Jenks criticized manager Ozzie Guillen to mlb.com. Guillen's son Oney, a former employee of the club, countered with a barrage of Twitter messages meant to defend his father that went on to describe personal incidents that happened in the confines of the White Sox clubhouse.
Oney Guillen also was critical of general manager Kenny Williams via Twitter messages last season.
"Me and [bench coach] Joey [Cora] talked about it and agree, and he's talked to Ozzie about the same stuff," said Beckham, who took part this week in a mini-camp for White Sox players hosted by Cora in Florida. "We will concentrate totally on baseball.
"We don't want excuses that cause us strain to win games. We want it to be on us, the players."
Paul Konerko, who re-signed with the White Sox, said off-field distractions can take away from the team's goal of winning the division. Matt Thornton said Oney Guillen's tweetings have to be "addressed quickly and taken care of and snuffed out real fast."
Without saying that he has heard Oney Guillen will be less revealing on his popular Twitter account, Beckham seemed to imply that private White Sox matters will now remain just that -- private.
"It's unfortunate that a lot of stuff came out, but it's over with and done, and hopefully we can concentrate on baseball," he said. "That's what will happen, I guarantee it."
Jenks commented on his former manager shortly after he signed with the Boston Red Sox in late December, saying, among other things: "I'm looking forward to playing for a manager who knows how to run a bullpen."
Among the Twitter claims that followed from Oney Guillen were that Jenks was having trouble with alcohol, cried in the manager's office about his "problems," diagnosed his own injury last season so he wouldn't have to pitch and assaulted a clubhouse attendant in Arizona.
While Beckham's comments came when he was asked about Oney Guillen's Twitter messages, there were other distractions last season. Chief among those was the strained relationship between Guillen and Williams that wasn't mended until the last week of the regular season.
"I think everybody is on the same page," Beckham said. "The players and coaches want to compete and play. We don't want outside drama [to] fill our locker room this year."
Ozzie Guillen has stayed quiet on the matter this offseason and Cora did not return a phone call Thursday seeking comment.
Doug Padilla covers the White Sox for ESPNChicago.com and ESPN 1000.