The Buddhist angle of achieving this seems to be to quiet your mind, ushering in a change of perception in your consciousness in order to care less about something stressful. To focus on the peace of a quieted mind. The Christian angle–which acknowledges an Omnipotent, Sovereign, and personal God–would have you take on the action of letting go in a kind of offering to Him. In a sense, offering the feeling of control we think we have, rightly back to God. So, instead of “letting go” and “detaching one’s self from a result,” the Christian sequence would read more like “Trusting God” and “desiring to live within His will for your life.”

As a Christian (if you couldn’t tell already ;) ), I think there is definitely a place for a practice of quieting one’s mind and letting go. However, if the ‘letting go’ is the only place the practice leads, I’d fear the loss of passion and ambition, not to mention the dulling of the sense of justice, for one. When we seek and ask things of God from within in our quieted minds, he will instill in us the discernment for what to care about and work on, and what not to worry about at all. He loves us, and loves when we earnestly seek peace through Him.