Illinois, with 102 counties, seems to be a bit more challenging than the ~1500 miles it takes to get to the
15 county seats in Arizona.
I cheated and used a route optimizer I found on-line that gave me a route. No, I've yet to accomplish it - but, it's on a list of 'someday' rides.
https://en.wikipedia.org/wiki/List_of_counties_in_Illinois
If the goal is to simply hit all 102 locations, get a picture and move on to the next, over a certain period of time ( a week? a weekend? a month?) the zig-zag routing that's used for many of the recent 48 state rides would seem to be efficient.
A longer period of time (a month?) might have me take smaller stabs, basically by quadrant. HItting 20 or 25 in a weekend could be routed fairly easily - and possibly accomplished in a weekend.
...and, it's this very type of puzzle that gets people into rally riding!
EDIT: Ran across this, seconds after navigating away...
https://www.mapquest.com/routeplanner
Find a list of the addresses, put them into this & see what happens!