I don't think we can expect to have one general answer to this question, only a collection of unrelated specialized results. Here are two more:
- In the category of simplicial sets all filtered colimits are homotopy colimits.
- Splittings of idempotents, which are colimits over the category freely generated by one idempotent, are always homotopy colimits.
On the other hand, your example with homotopy pushouts does fit into the general framework of cofibrant diagrams. There may be more than one notion of cofibrant diagram suitable for deriving the colimit functor. In this case, if we denote the objects of the indexing category as $a_0 \leftarrow a_1 \to a_2$ and make it into a Reedy category by declaring the degree of $a_i$ to be $i$, then Reedy cofibrant diagrams are spans where one leg is a cofibration and the colimit functor is a Quillen functor with respect to the resulting Reedy model structure. Perhaps Gregory's example with cubes can also be described in a similar manner, but it is less obvious to me.