I know some people have had success using the Matlab code
iso2mesh, which is designed to create a tetrahedral volume mesh from a set of isosurfaces (the surface mesh you can get easily within DREAM.3D). That may be a reasonable place to start. I've also found
this webpage helpful in the past for exploring some of the available options around for creating solid unstructured meshes. Some of the packages listed there are out of date, but others are under active development.