To be a living sacrifice is that we live counter culturally.  When we present ourselves as a living sacrifice, it transforms us and we begin to change our thinking. The thinking of the surrounding culture is based on another worldview with entirely different assumptions.  This is that to which we do not and cannot conform.  Our thinking is changed because of what Jesus has done for us; it has changed because of what He is doing in us.

“And so, dear brothers and sisters, I plead with you to give your bodies to God because of all he has done for you. Let them be a living and holy sacrifice—the kind he will find acceptable. This is truly the way to worship him.  Don’t copy the behavior and customs of this world, but let God transform you into a new person by changing the way you think. Then you will learn to know God’s will for you, which is good and pleasing and perfect.” Romans 12:1-2 (NLT)