Except that the constitution isn't quite as explicit as you apparently think, because there is no categorical prohibition on warrantless searches in the constitution. The fourth ammendment reads,
"The right of the people to be secure in their persons, houses, papers, and effects, against unreasonable searches and seizures, shall not be violated, and no Warrants shall issue, but upon probable cause, supported by Oath or affirmation, and particularly describing the place to be searched, and the persons or things to be seized."
So we have a probable cause requirement for warrants, and we have protection against unreasonable search and seizure. But we do NOT have any constitutional requirement for warrants for all searches. Nor, in fact, should we, as a few moments consideration will show. For example, let's say a policeman hears screams for help, followed by silence, from a house. He may choose to search the house, without a warrant, because the search is not unreasonable given the particular circumstances. Waiting for a warrant in such a case is not reasonable. So there is no categorical requirement for warrants in all cases, there have always been exceptions and there's nothing unconstitutional or improper about having exceptions.
I do not know enough about the particulars of these NSA wiretaps to know if they were legal, and I do not know enough about the particular laws involved to know whether, even given their legality, they SHOULD be legal (because certainly it's possible to make the exceptions to warrants too broad). But I suspect that the matter is likely not quite so black and white.