Secular Moral Compass

Last night while doing my quite time I asked God to stretch my brain in the reading of his word. I was reading Luke 6:27-36. It talks about loving your enemies and praying for those who persecute you and how loving those who love you is easy and of no credit to you. 
there was one part that stood out to me though "Do unto others what you would have them do unto you". Now I'm fairly certain that this does not need to be explained, it's 'The Golden Rule' the one that has been ingrained in us since childhood, simple and effective. What I think is interesting thought, is, that society has forgotten that this verse has it's root in the bible, it's directly quoted from Christ himself.  Why do we accept some parts of the bible as fact and incorporate them as part of of our societal values but will not accept the whole thing? If you accept the bible's teaching on how we should treat each other, why not accept the whole thing?

As a Christian, I believe, that it is wrong to lie, steal, cheat, murder etc (Please do not assume that I am implying that Christians are 'morally  superior' or that non-Christians can't hold these views.)
The reason I believe these things, is that God created all things and created us for relationships; both between us and God and each other. These things; lying, cheating, stealing, murder etc break and ruin these relationships and it wrecks how we were created to be. (I am not questioning where or not these things are wrong, but what moral basis does or should this be tied to?)

For a largely secular society to adhere to the Christian standards of 'morality' seem a little bit backwards when it often rejects Christian teaching as a whole. And if the bible support for these things is rejected what then is the basis of thinking that these things are wrong? 

I guess what I am trying to say is: What does society as a whole, base it's moral compass on, if it is not the bible?
Why do non-Christians believe that murder is wrong, that lying is wrong, that stealing, that cheating is wrong? What substantiates that?

Where does society base the concept that murder and lying and cheating is wrong? It is our laws or governments, where do they get the authority to make these things fact?

Let's just make the assumption that our society's moral values are based on christian principles. However, today the traditional christian opinions that helped shape our society are being questioned. Views on homosexuality, marriage and abortion have changed and the traditional christian view point on these things is being challenged. What is to stop the traditional views on murder and lying and cheating being changed? What is to stop them from becoming socially acceptable, because the majority of society has rejected christian teaching. If they have no solid grounding in some form of morality what is to stop these things from happening?

I am of the opinion that we as a society think that murder is wrong because the bible says it's wrong. What about atheists and agnostics that reject all of the Christian teaching, shouldn't they then also reject what the bible says about lying and stealing and cheating, would there not be totally anarchy? I am curious then, what do they base their moral compass on?

I'm concerned that if the basis of society's views on what is right and what is wrong, and what is acceptable and what is not acceptable are not grounded on something bigger than just this human consciousness, of what is good and what is bad, we are just going to lose grip.  Where's the line?

As a Christian I believe that society's views on morality should be grounded in the bible.