The English run Speedway bikes on grass sometimes - well, it starts out as grass before a bunch of 60 hp alcohol-burning Jawas run over it. Agree a grass track doesn't quite look like the San Hose Mile - but if it was flat enough, the grass wouldn't last long, I'll bet.
Edit: Ah, as it turns out what I thought were speedway bikes on grasstracks actually turn out to be the very similar - but different - grasstrack bikes, which have transmissions and a few more classes, including 2-strokes. Anyway, if the Angles and Saxons can do it on the grass, why not here in the colonies?