I've been pondering what the text of a proposed amendment disallowing routine DNA search would look like, and here's a shot at it:
No person shall be compelled to yield organic materials from their person except with the case of probable cause, supported by Oath or affirmation, and particularly describing the crime under which the person has been accused.
I think this places a higher burden of proof, and also draws a bright line as to what we consider to be a more intimate violation of the fourth and fifth amendments. I wasn't able to find any proposals floating around online, but I suspect that this is the type of thing that would probably have a reasonable following.