For those finding this thread on Google...I had a similar problem with my Kenmore 587.1525 dishwasher. I could start the wash, the washer would fill but the rinse cycle wouldn't initiate.

For me it ended up that the control board wasn't sending the spray/rinse signal to the motor. I decided to reboot the washer so I cut the power (in my case at the circuit breaker) and waited for a few minutes. When i flipped the power back on two of the front panel lights flashed. I pressed cancel and started a new wash. Thankfully everything worked correctly.

Cutting the power also fixed a problem where the panel was completely unresponsive.