Mindfulness is the phrase of the moment. Even in some Christian circles. Even, I might add in my Baptist church (the church I'm part of at the moment, I should clarify). The teacher the week before last extolled 'mindfulness' as a practice for Christians.
Mindfulness is Buddhism cleaned up for we Westerners.
But Christians have their own tradition of stillness, contemplation and quiet. It stems from the Psalms where David was consciously in the presence of God, preoccupied with the word of God, not how relaxed his toes were.
I would expect Christians to be engaged people and aware of who is around them, and the emotional climate they are in...of course we can't do this all the time. But I would also expect that Christians take time to be quietly alone with our heavenly Father in prayer and meditation on his word. This is enjoined by the Scriptures.
Similar is encouraged in an article in the Melbourne Anglican.
A Christian New Age info site has more detail.