Ken Brown (of Brown Representability fame) published a paper relating Cech hypercohomology and and sheaf cohomology, extending a result of Verdier (that is relegated to an appendix in, I believe, SGA 4. Brown calls it "Verdier's Hypercovering theorem"). There is an evident way to compare Cech cohomology and Cech hypercohomology (as discussed in the paper), and so I believe that it should answer your question:
Edit: If the link does not work, the paper is Abstract Homotopy Theory and Generalized Sheaf Cohomology by K.S. Brown