Mt. Pisgah in Eugene is a great place to take dogs and go hiking. For an easier hike, look for the arboretum trails. If the weather is right there are some creeks in the parking area that a dog could play in.
Tryon Creek Trail near Portland is dog-friendly and just friendly, in general.
Wahclella Falls near the Columbia River Gorge is an easy waterfall hike that is dog-friendly.
The coast is very dog-friendly with lots of hikes and even yurts that pets can stay in with you.
Cape Lookout South Trail near Tillamook is a great example of the Oregon Coast’s wide array of dog-friendly hiking trails.
Cape Perpetua’s Giant Spruce Trail: This trail is one of my favorites on the coast.
Eagle Cap Wilderness: Hurricane Creek near Joseph is a dog-friendly hike in the northeast of Oregon.
This book is a really good reference for hikes with dogs: http://www.mountaineersbooks.org/Best-Hikes-with-Dogs-Oregon-2nd-Edition-P705.aspx
I hope this helps! Happy travels!