Dishwasher backs up in sink

I have an old Kenmore dishwasher and every time it empties water comes up into the right sink (which has a garbage disposal). We used to wash our dishes in this sink but now we use the left sink so no water comes up into the standing water. Any ideas on how to stop this? We just replaced all the pipes and the disposal under the sink.

If only one sink is clogged, the clog is isolated where the garbage disposal is. Take the piping apart and locate the clog and remove it.