A common argument one hears about the quality of the United State's government and God is that the "In God We Trust" motto found on US currency only started being printed under the Eisenhower administration to make a statement against the godless communist atheists of the USSR during the cold war in the 50s.
While that does have a bit of truth in it, it was only first printed on paper money in 1957 and it has only been the national motto since 1956, but the fact is that the phrase "In God We Trust" has been used in US coins since 1864 due to issues raised by the Civil War and worries that future generations might think the US a heathen nation if she were to be destroyed and all that was left for future archeologists to discover was the godless coinage. See here.
And even if we were to ignore the blatant "In God We Trust" motto, why should we ignore the "Annuit Cœptis," latin for "He [God] has approved our beginnings" which is on reverse of The Great Seal of the United States, which has been on the one dollar bill since 1935 and which has been used as the seal of the United States since 1782. Not to mention the all seeing eye of Providence which appears above the unfinished pyramid right below that phrase.
The truth of the matter is that the 1950s was not a break in America's character at all, but actually a continuation of God being a part of the national character since basically its inception. There are those atheists who argue that such national expressions of God are actually religious declarations of religion and are therefore unconstitutional, but I think they'll have a hell of a difficult time proving that given the significant historical connection between the United States and God. Though, of course, no sign of what kind of God it is, but still it would appear that some kinds of theistic notions are implied.
Will the trusting in God by the government ever be deemed unconstitutional? Maybe. But why bother? It would be far better to secularize the idea and make the God of Providence and the one in whom "we trust" into the same sort of deity as the Goddess of Liberty found currently en large in the New York Harbor. Why reject what we can always incorporate?