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:

[Click!](http://ncatlab.org/nlab/files/BrownAbstractHomotopyTheory.pdf)