I'm not sure if I know what you mean by "easy alpine routes," but if you want remote, the North Cascades is a great place. Some not-to-technically alpine routes that are wonderful in the North Cascades are the Sulphide Glacier on Mt. Shuksan, Sahale, Eldorado Peak, Silverstar Mtn. not to mention Glacier Peak and Mt. Baker, and many others. In the central Cascades, it might be worthwhile trying to get into the Enchantments. Of course, there is always Mt. Rainier, which has moderately easy (though technical) routes.
If you're looking for non-technical climbs, i.e. scrambles, look at Peggy Goldman's Book on 75 Scrambles in Washington. Also, Jeff Smoot's book on Climbing in Washington is a good resource.
Other remote areas are the Paysaten Wilderness and the Chelan-Sawtooths area.