I would try a variation on the XXX team ride route.

Custer S => Howard
Howard W => Ridge
Ridge S => Ravenswood
Ravenswood S => Bryn Mawr
Bryn Mawr W => Damen
Damen S => Clybourn
Clyboun S => Fullerton

Your dodgiest section will be Ridge. Kind of narrow and busy, but doable. You could try any variety of side streets if you decide the traffic is too much.

Damen has a bike lane most of the way. You still need to be on guard against doors, but Damen is my primary north-south artery (not counting the lake).

Ravenswood is also very bike-friendly, but you'll have to do some navigation to get all the way to Fullerton.