I would submit that we have no rights, at least not in the eternal sense. To argue otherwise is to adopt an entitlement mentality. Sadly, we Americans can't seem to help looking at liberty from the perspective of what we believe we are entitled to. When we complain about government intrusion into our lives, we gripe about "our freedoms" being taken away.
We need to remember that everything we have was given to us. We are entitled to nothing. In fact, the only things we deserve are death and eternal punishment (Romans 3:23, Romans 6:23, 2 Thessalonians 1:9). So, from a Christian standpoint, how do we reconcile our theology with the traditional American view of liberty?
The best way to interpret the concept of "inalienable rights" is to flip it around. When we speak of the right to life, liberty, property, the pursuit of happiness, etc., think of it not in terms of what a person deserves, but what a person isn't entitled to do to someone else. It doesn't mean that I am entitled to life, since that is an undeserved gift from my Creator; rather, it means that you have no right to take my life. The Bible calls that murder (Exodus 20:13). You have no right to take my property. That would be stealing (Exodus 20:15). And so on.
One important thing to note about the Ten Commandments is that in using words like "you shall not," they focus on God's rights, not ours. Since all men are created in the image of God (Genesis 1:27), all men should be treated with dignity, respect, and love -- not because they deserve it, but because God deserves it.