Where in downtown Portland, OR can I rent lockers for my backpack?

The Greyhound bus station and the train station both used to have short term lockers, but they would charge each time they were relocked. Since September 11, 2001, lockers have largely disappeared (I suppose) for their potential to house terrorist devices.

If it were me, I would leave my backpack at one of the nicer hotels (The Nines, Heathman, Modera, Monaco, Lucia, Ace, Vintage Plaza, Hilton, Marriott). They all provide a luggage holding service for departing guests after checkout but before the guest departs to the airport (or wherever), so their guests can sightsee or do business on the last day without their luggage. Usually there is no charge. Offer to pay, but since they don't charge and wouldn't know what price or how to handle a fee, they will probably just take it for free. (They do accept tips though.)