I live in Portland and ride out to the Gorge regularly. Like Jennifer said, there are pros and cons to each. But I'd recommend the Oregon side for a few reasons:
- It's got the better highlights and scenery. You'll get some great viewpoints like Women's Forum/Crown Point, the loops from Hood River to Mosier, and Rowena Crest. Plus, there's all the waterfalls. If you do the Washington side, you'll get one great view at Cape Horn and more overall views of the river.
- There are some pretty nice car-free segments like the one Jennifer mentioned, plus those Mosier loops. They keep on adding more car-free sections on the Oregon side. The Washington side has nothing.
There are definitely cons to the Oregon side, like the sections of I-84 shoulder that you still have to ride for now.* Plus, there's a set of unavoidable** stairs near Eagle Creek that will be a pain if you have a recumbent/tandem/trailer/etc. But WA 14 is so busy nowadays, and there isn't a lot of shoulder. I rode a section of 14 between Beacon Rock and Bridge of the Gods last weekend, and it was pretty hairy, with lots of big trucks. I'll do 14 if I need to, but not by choice.
WA 14 is pretty good east of The Dalles, with a nice wide shoulder for the most part. So you can do Oregon side from Portland to The Dalles, then 14 east of there.
*Wyeth Bench Road between Cascade Locks and Wyeth is pretty steep, so I'll probably keep on riding the shoulder of 84 in that area.
**The only way to avoid it is by getting on 84 as it goes through shoulderless Toothrock Tunnel. Or hopping on the bus.