You might want to take a bus or drive to reach it, but the Burke Gilman trail could be nice - to the west if you want to take in the city sights, to the east if you want more of a suburban experience. You'd have the option of running on the pavement (it is a multi use trail so be mindful of cyclists) or along side on the dirt in many places.

Here's a good trails resource - this is for King County, so it goes fairly far afield, but most of the trails, even in the city, are "county parks" - just really long skinny ones and the map should be useful to you.

http://www.metrokc.gov/parks/trails/