As you say, most natural hydrogen is not radioactive (i.e. excluding a small natural tritium content. But, during a fusion reaction both ionising and non-ionising radiation (e.g. neutrons, gamma rays, x-rays and alpha and beta particles) will be produced. Some radioactive isotopes (e.g. tritium) will also result from the reaction.
More signifcantly, if a fusion bomb is exploded close enough to ground level a significant quantity of surface soil, water, plants etc.. will be lifted and activated by the intense radiation. This can give a very wide variety of radioactive isotopes which will be manifest as 'fallout'.
Tests have shown that a fusion bomb exploded high enough produces less fallout, but syphrum says, the fission trigger can be significant on its own.