There is evidence some plants are better suited to certain heavy isotopes. Sunflowers, for example, were shown to draw cesium into their stalks [1]. The contaminated stalks would then be gathered and destroyed in a controlled manner to extract the radioactive ash.
However, in the case of Fukushima, the millions of sunflowers grown after the disaster started didn't seem to do much at all. [2]
Plants don't help when we choose to incorporate their radiation into our bodies: Tobacco pulls radioactive polonium-210 from the soil and air[0]. It has been found to accumulate in the lungs and continue to emit alpha-particles.
Just a note, if you've only seen the movie but not read the graphic novel it was based on, try to find a copy! The movie only covers about the first 1/4 of the novel, which is more expansive thematically as well as in plot. In particular, the ending of the novel is much more ambivalent than the mostly simplistic moral of the movie.
Thank you, I'll add that to my list of things-to-read.
I have the movie on DVD and on the bonus material it was said that Miyazaki wanted to make the movie first. At that time anime that wasn't based on a graphic novel (or novels) was practically impossible to get into wide distribution. Hence, no funding was available.
So they started by writing the story in a form of graphic novel and had it published first. After a moderate success in manga form it was easier to get the deals for making the anime too.
The bonus material went far enough to state that without the film there wouldn't be Studio Ghibli. Personally I love the film. Even among the Ghibli productions it stands out as a great work of art.
Bury them deep underground. An deep abandoned mine would be good. They are not so radioactive you need serious security, but they do need to leave the environment for a while.
Burn them in an isolated manner to reduce volume/weight; and bury the ash as (mildly) radioactive waste - thus removing the harmful isotopes from a lifecycle that might end up in humans.
I'm not surprised that slower growing vegetation is showing an uptick.