Between 6AM and 10AM is the optimal time. Before this and you risk fungal growth. After this and the sun will evaporate most of the water away.
Kansas City metro has clay loam soil. This means that the soil holds the water very well. This is great for grass as it encourages deep root growth necessary for lush green grass. That being said, you should water deeply, but infrequently. Your sprinklers should put between 1″ and 1.5″ of water on the ground each session. Even at the height of the summer grass only needs .25″ each day to be healthy. That means in the middle of summer, your sprinklers should only run twice per week. In the fall and spring no more than once per week. The key is to water deep and in the morning so the soil soaks up all the water.
The more you mow, the better your grass will be. Mowing encourages new growth and healthy lawns should be mowed twice per week. Make sure your blades are sharp, if not, they will beat the grass and cause harm as opposed to cutting it.