The Republicans' Pledge to America states that they should. Andrew Sullivan highlights dispute concerning the requirement:
The passage makes Ben Adler of Newsweek uncomfortable: "Not so harmless, however, is the promise to require every bill to be certified as constitutional before it is voted on. We have a mechanism for assessing the constitutionality of legislation, which is the independent judiciary. An extraconstitutional attempt to limit the powers of Congress is dangerous even as a mere suggestion, and it constitutes an encroachment on the judiciary."
Ramesh Ponnuru cries foul: "There's nothing -- nothing in the Constitution, nothing in Marbury v. Madison, nothing even in Cooper v. Aaron -- that suggests that congressmen cannot consider the constitutionality of laws while voting on them. That they can do so, which one would have hoped would be a banal idea, does not even challenge judicial supremacy: The courts can still be the final arbiter of constitutionality. The Pledge provision in question is "extraconstitutional" only in the trivial sense that the Constitution neither requires nor forbids it. And the provision is better understood as a spur to congressional self-restraint than as a "limit" on congressional power."
My two cents: It's a great idea. All laws have to have a basis in the Constitution anyway, so why not establish this up front? Supreme Court decisions often begin with language that explicitly links the legislation to the relevant part of the Constitution. Supporters of a given law should actually be anxious to do so because instead of relying on legal reasoning fabricated by litigants to a suit regarding the law (people who may have had nothing to do with the making of the law), they can pre-determine what argument ought to be made to the court. They can, perhaps, cut off constitutional challenges before they get legs. Judges and justices would not be able to claim that they could not determine the constitutional intent of the law, because that would be made clear in the law's language. It is, tactically, a great idea.
It also serves a greater purpose. It contributes to civic education by fostering a more intelligent conversation about the Constitution and how it should be interpreted. For those who argue for a broader interpretation of constitutional language, it would also provide a stronger platform to argue that the legislation is in fact based on a reasonable interpretation of constitutional language, even if it is not always convincing to opponents. This could (could, I don't want to be naive) help raise the level of discourse about public affairs. Democrats should support this proposal (it would be so post-partisan to do so). By not engaging in this conversation, they simply help strengthen the Tea Party position that they, and they only, speak for the Constitution.