I first heard the phrase "winning hearts and minds" around the time of the U.S. invasion of Iraq. I have no idea if it was used before then, but lately it's been a reference to the effort the U.S. military would use to gain the support and trust of the Iraqi people. I'm inferring that the plan involves convincing the Iraqi citizenry (minds) we're right while also becoming their pals (hearts).
Unlike "shock and awe," the media is still beating to death "hearts and minds." Richard Engel, an international correspondent for NBC news uses the phrase weekly when reporting on Iraq. Newsweek journalists used it in their September 17, 2007 article entitled "Braniac Brigade." I'm sure the rest of them use it too, but I'm only catching the stuff I watch or read regularly.
It's ubiquitous and persistent and needs to be stricken from the English lexicon immediately. Richard and his peers are intelligent people; I know they know better. So for the love of god people, find a different way of describing the effort to win the support of Iraqi citizens. Because if I hear this phrase one more time, I might just have to rip out my own heart and mind. The U.S. Census Bureau could tally my suicide as "Death by Triteness."