The Declaration of Independence was as much influenced by religion (not entirely Christian) as most other documents from the time. There were fewer outright atheists as there are today; most people accepted a basic religious belief, and so would pepper their words with references to religion and God without really thinking about it. For the Declaration, it was more about establishing the rights of man to exist beyond the control of a mortal king than it was about trumpeting religious beliefs. Since most people at the time believed in a divine creator, it was natural to compare that creator with the king, who the Founders saw as being immorally oppressive; a man should never have that kind of power over other men.
I would say -- without doing extreme research -- that influence over founding documents is less explicitly Christian and more explicitly philosophically moral from a religious standpoint. In other words, the morals people had came from their religious beliefs, and so it was natural to use those norms when writing. Today, natural philosophy has progressed to a point where we don't need to ascribe our morals to a divine creator, and so we can simply ignore the religious subtext and focus on the fundamentals and the substance of the documents.
Given my peeked interest in this question, I pulled up "The Declaration of Independence." I found two lines spoke directly to what one could assume to be Christian influence: "Laws of Nature and of Nature's God entitle them" and "they are endowed by their Creator with certain unalienable Rights." Given that God in not directly stated, many must assume that "Nature's God" and the "Creator" are God in a Christian sense. Therefore, I would agree that there are "traces" of Christian influence, but only if one acknowledges the traces to be Christian.
The influences on the Declaration of Independence are not just Christian. The founders were also influenced by Greeks, John Locke, and other philosophers. The concept of unalienable rights is often attributed to influence by John Locke.
In his work, the combination of natural law theory and the concept of vested rights was clearly stated for the first time, transforming the latter into the principle of inalienable rights. (see first link)
Jefferson clearly approved of the idea. The unalienable rights of life, liberty, and the pursuit of happiness were the foundation of the Declaration of Independent and because the biggest influence in the new concept of the American Dream.
This does not mean there was no religious influence in Jefferson. By all accounts, his religious views differed from others of his time. He strongly opposed the prescense of religion in government.
I'm not sure what the original question meant by "traditional Christianity," since most evangelical faiths today have their roots in movements of the eighteenth century that were anything but traditional at the time the Declaration was written. In any case, I think the previous response is exactly right. Specifically, the other religious references that can be found in the Declaration are vague, referencing "Nature's God," "Divine Providence," and a "Creator" that endowed men with the unalienable rights. It is also worth pointing out, however, as Pauline Maier does in American Scripture, her book on the writing of the Declaration, that Congress actually added most of these references to God after Jefferson submitted his draft. The only reference in the original draft was the one to the "Creator." So there was some intent by Congress to sacralize the document by adding a few references to a divine being, however vague.
Source: Pauline Maier, American Scripture: Making the Declaration of Independence (New York: Vintage Books, 1996) 148.
This really depends on what you mean by “Christian influence.”
There are many people who argue that the whole idea of natural rights is a Christian idea. They say that the idea that all people are created equal and the idea that all people have these basic rights comes from Christianity. If this is so, then that is Christian influence. But there is nothing in those two ideas that is inherently Christian. It is very possible for people of other religions or no religion to hold those ideas.
As far as explicit references to Christianity, they just aren’t there. Yes, people are “endowed by their Creator” with their rights. But nothing says that that “Creator” has to be the Christian God. The other references to God are similar in that they could be Christian but aren’t necessarily.
Thus, there really isn’t anything in the document that could only come from Christianity.