Yup. Skykomish and Wenatchee are fairly easy and/or have immediate road access.
The White Salmon also has a nice class III section.
The Wenatchee, as Kiwi pointed out, is a super friendly big-water style run. Doesn't have a single rock or boulder in it to speak of, just several good play spots, depending on Waterlevel.
The Sky, is a great pool drop class III run with one class IV called Boulder Drop -one of the most popular runs in the state. B.D. can be scouted easily and portaged as well.
The Green Gorge is considered class III, but it has a remote feel- very scenic for sure, but also borderline class IV in several spots,and kinda manky in places. That one is a judgement call you'll have to make.
Check out Jeff Bennett's "Guide to the Whitewater Rivers of Washington", or go to
http://www.americanwhitewater.org/ and look at Washington's runs for the descriptions they list there.
When you get here, you can also use this site to hook up with lots of different people to paddle with. Just remember that it's your personal responsibility to research any and all runs ahead of time for yourself, and to look out for youself and your own anytime you get your feet, boat, paddle or head wet.Having said that, Have fun up here!
This region is the best place to be a paddler, IMO (I'm a transplant).
You can also check out a company called Wavetrek- they offer swiftwater rescue courses, and probably kayak instruction as well.
Edited by wetmouse - 27 May 2009 at 6:44pm