When people speak of Ancient Greek Democracy they often fail to forget a key element, that is sortition. In Athens sortition was considered far more fair than elections and allowed true rule by the people as opposed to oligarchy.
So, what is sortition? Essentially it is how jury duty is supposed to work. It is a simple lottery or other method of random selection that appoints people to positions.
In this case, why not use sortition to form government. We could start at the local level where turnout is absolutely abysmal anyway. Of course, while turnout is absymal for local elections, people fail to realize that these individuals have more effect on your daily life than the president ever will. So while Jim Bob Smith gets elected police commissioner and his cousin gets elected County Treasurer for the seventh time in a row, maybe it is time to think of a new way to run things.
Sortition.
Have every region divided into Local Government Councils and all councilmembers appointed through sortition. Of course age minimums and health concerns would be taken into account. People currently in prison may have to be excluded as well. But that can all be worked out. Otherwise, why not?
Seems far more representative than elections.