How to create fruity or grassy shortbread cookies?
Three things:

-You can flavor the sugar by mixing aromatic herbs into it and leaving it sit for a week or two. Rosemary pairs well with sweet desserts, as would lavender or basil. Sift out the herbs once you're ready to use the sugar.

-You can try replacing the butter in this recipe (or half of it) with some very good olive oil, as in this recipe from the NYT:

http://dinersjournal.blogs.nytimes.com/2008/03/07/recipe-of-the-day-olive-oil-cookies-with-red-wine-and-rosemary/

That recipe isn't for shortbread cookies, but the technique may still be valid. No guarantees that it will give you the texture you want, but great olive oil has all of the grassy and fruity characteristics that you're after.

-I'm not terribly familiar with matcha, but couldn't you just take some regular green tea leaves and powderize them in a spice grinder?