The problem with Cech cohomology is that even if things are acyclic on open sets of your Cech cover, they may not be when you restrict to intersections of those open sets. The usual fix is to make the cover finer so you don't have that problem. Unfortunately there are topological spaces where no cover will be good enough. That's the bad news.
The good news is that for a lot of spaces and categories of sheaves you're interested in, there will be such a cover. My favorite example is the category of quasi-coherent sheaves on a (noetherian) separated scheme. Then Cech cohomology computed on any affine cover will compute the derived functor cohomology.
The even better news is that there is a way to fix Cech cohomology so that it will work for all situations. This is Verdier's theory of hypercovers, and it computes derived functor cohomology for any category with a Grothendieck topology. I must admit I have not played around much with this, but here is a link to a paper that talks about this circle of ideas.
