Not sure it's the best way, but this is the route I took:
1. Ne-50 South to US-136 (in Tecumseh)
2. US-136 to US-75 South (in Auburn)
3. US-75 to US-73
4. US-73 to US-36
5. US-36 to I-229
6. I-229 to I-29
I took this route for several reasons:
1. Obviously I-29 being closed north of Rock Port
2. Construction on US-75 between Omaha and Nebraska City
3. Bridge down to one lane on US-136 in Brownville
4. US-159 closed east of Rulo
Drive time was 3h40m, leaving from west Omaha to Olathe. There was no construction and very little traffic until US-36 (that's where you'll run into dumb Missouri and JoCo KS drivers who can't drive a two lane highway). Speed limits were either 60 or 65 mph.