The agitation cycle of my washing machine causes all my lights to flicker

First off, please remove your CAPSLOCK.

Second, when you say flicker, do you mean literally on and off, or are they just fluctuating slightly brighter and dimmer?

Indicates a voltage drop issue, usually due to a loose or poor connection, could be at the utility connections and/or in your panel.

Best to call in an electrician to troubleshoot the issue.