I've read about this concept in the Riverworld series.
It doesn't end well for them.
Random DNA sprayed around a crime scene would just seem weird. Wouldn't they also have to make it seem real? IE fake blood, saliva, whatever, in the same places it would be found in a crime scene?
"Can't you just gesture hypnotically and make him disappear?" "It does not work that way. RUN!" -- Hadji on metaphyics and Mandrake in "Johnny Quest"