Unless the unpainted walls substantially endanger or impair the health or safety of the tenant, the landlord would not be required to paint them for a new tenant under the WA Chapter 59.18 RCW (Residential landlord-tenant act). The statutes are at http://apps.leg.wa.gov/RCW/default.aspx?cite=59.18 .
The local governments (city or county) may have ordinance requiring the painting.
Alternatively, your rental agreement may require the landlord to paint the walls. You should review your rental agreement to see what each side agrees to do.
You should record by video or photographs the condition of the rental when you first move in so that you have evidence should the landlord later charges you for pre-existing conditions.