Where those weapons get the most energy from depends heavily on the design. It's true that multi-stage thermonuclear devices often have a final fission stage where huge amounts of energy are released, however if one were to purposefully maximize energy release while minimizing fallout there are some relatively clean fusion/fission ratios achievable.

For instance, the test version of the Tsar bomb had its last fission stage removed, reportedly resulting in a 97% fusion detonation. Of course, those still cause a huge lethal neutron flux, but at least the amount of fission material released is pretty small in relation to the overall power of the device.