Torsten Ekedahl proposed a definition of higher Grothendieck groups of varieties. Unfortunately it seems that he never wrote anything down on this topic before passing away. Torsten had quite a large number of unfinished mathematical manuscripts and projects. I don't know what happened to them, although surely someone at Stockholm University has taken care of them. 

I saw him give a talk about higher Grothendieck groups of varieties at Gerard van der Geer's birthday conference on Schiermonnikoog in 2010.  The abstract is available online: http://www-irm.mathematik.hu-berlin.de/~ortega/schierm/

> *“Higher Grothendieck groups of varieties”* 
>
> We shall (slightly) modify the setup of Waldhausen’s definition of higher
K-theory in order to introduce higher Grothendieck groups of varieties such
that the 0’th group is the ordinary Grothendieck group of varieties. We
then establish some basic properties of these groups that generalise basic
properties for the ordinary Grothendieck groups.