It was at that point that his wife suggested he create a Twitter account, allowing the two an easy -- and, more important, less expensive -- form of communication.
"I didn't have any idea what [Twitter] was," Thompson said.
In addition to sending tweets, users can "DM" (direct message) others, with DMs essentially replacing text messages.
"Texts were going to be ridiculously expensive," Thompson said, "so it was like free instant messaging -- that was the only reason why I got it."
Thompson still uses Twitter for the same reasons he did two years ago, keeping up with family and friends in Australia, but as Twitter's popularity has swelled, he's learned to value its other uses, too.
"Being an international player, and there not being a lot of coverage of baseball back home, it kind of gives an outlet for people to be able to be like, 'Oh, I wonder what he's doing today,'" he said. "You can just follow somebody. It's information straight to your phone."
The 26-year-old finds certain information a lot more enjoyable -- particularly when it's about automobiles, as he's a car fanatic.
"I follow a lot of car blogs," he said. "Pretty much all automobile news coming out in the world comes straight to me instead of [me] having to search 20 million sites."
And although plenty of fans follow their favorite players daily, athletes are often fans as well. Thompson used Twitter to meet an Australian cricket player he likely wouldn't have been able to come in contact with otherwise.
"He came out to L.A., and we got connected through that," Thompson said. "So with a lot of the athletes, it's almost like having a big Rolodex of everybody. If you want to try to get a hold of somebody whose number you don't have, you can."