My understanding of the general Mayer-Vietoris principle is as follows. We want to compute the cohomology of some sheaf $\mathscr{F}$. We start by taking a resolution $$\mathscr{F}_0 \rightarrow \mathscr{F}_1 \rightarrow \mathscr{F}_2 \rightarrow \cdots$$ of $\mathscr{F}$ by acyclic sheaves, then replace each $\mathscr{F}_i$ functorially by a resolution of its space of global sections in order to get a double complex, and finally construct the two resulting spectral sequences. One of these spectral sequences tells you that the total complex computes the cohomology of $\mathscr{F}$, whilst the other hopefully reduces to some simpler things you might already know about.
In practice (in topology, which is what I'm really interested in), $\mathscr{F}$ may be a (locally) constant sheaf, and the $\mathscr{F}_i$ the sheaves of singular or de Rham $i$-cochains, although of course one could consider much more general things. However, the resolutions of the global sections always seem to be of the same form, namely the Čech complex for some open cover (perhaps this is what is really meant by the name "Mayer-Vietoris").
Are there other resolutions that give useful spectral sequences? For instance, you could take the global sections of the Godement resolutions of the $\mathscr{F}_i$, but it's not clear to me that this would ever be a sensible thing to do.