I suggest the Chief Sealth Trail. I just discovered it yesterday, and I'm heartbroken I didn't spend more summer days there—it feels like a secret world tucked in the middle of the city. Starting in Beacon Hill, it transports you to Kubota Garden via Ireland. Or New Zealand. Or some other place with rolling green hills I've never been. Find the entrance like this: Head toward the north end of Beacon Hill (take 12th Avenue across the Jose Rizal Bridge—inhale that view), veer left around the Pacific Medical Center/Amazon (check out the weird new green lane—is it Astroturf?), take 15th Avenue South to Beacon Avenue, pass the Jefferson Park Golf Course (laugh at the street called Cheasty), and find the entrance just past South Columbian Way. (If you're confused, this bike map comes in handy. SDOT will even mail you one for free. So you can put it on your wall and gaze at it all winter long, planning your sunny-day adventures.)
The trail (paved, smooth) runs along Seattle City Light's utilities corridor, which provides for sharp intersecting (power) lines of sight as you ride up and down the hills (roller-coaster-style, so gravity does the work for your lazy legs) and catch glimpses of Lake Washington through the turning trees. Also: Peep Rainier Valley's monster-size P-Patches! (With sunflowers!) Also: Trip on the fact that you are riding atop the excavated soil and concrete from the light-rail construction along Martin Luther King Jr. Way.
And if you were to risk life and limb by listening to music while you ride (the horror! The danger!), Throw Me the Statue are the ideal accompaniment.