Fair trial w/ witnesses goes back in codified form to Hammurabi.
Private property rights existed before the 10 Commandments, and were officially codified ~ 2300 BCE in the kingdom of Lagash.
Judeo-Christian philosophy is heavily influenced by the land surrounding its infant stages: the various Babylonian and Persian societies. It is likely the concept of a "devil" for example comes from Persia; it wasn't in the Old Testament. These societies all held some of the same rights and privileges of social organization that we do now but it was not the exclusive province of Christians or Jews to define what those rights and privileges were.
Being practical people, it's easier to attract converts to your tribe if you have similar practices, or if you adopt practices which seem to work for other tribes pretty well. The idea that theft, rape or murder are bad doesn't require any God to tell us that. Simple logic suffices, and none of these things are expressly mentioned in the Constitution for this reason. Treason, libel and slander are in fact the only crimes mentioned. Libel and slander go back to Hammurabi in theory, probably again further back than that.
While the facts are that the Constitution was written by mostly Christians, that does not make it a divinely inspired document or a Christian document/endorsement. It's a secular document with careful attention to the power of the state to enforce upon unwilling minorities' rights. Laws are therefore to be made by humans, enforced by them, etc, not to be handed down from on high.
New issue of Econ Journal Watch
24 minutes ago