Going down the Columbia is scenic, esp. between Portland and The Dalles. However, there is a good stretch between Biggs and Umatilla that is less scenic, but still cool in its own way. April or May is the switchover from winter (where wind blows from the east) to summer (where it blows from west), so it's a toss-up. You might not face the stiff west wind that you would get in the middle of summer, so it might not be as bad.

The tough thing is getting out of Spokane, as there isn't a heck of a lot of anything going south and west. I rode Portland to Spokane a few years ago so I have familiarity with it. The problem with SE Washington isn't hills (though there are a few of those, but no mountains like most other regions of Washington), it's the lack of services. The area is heavily farmed/ranched (very little "wild" land) and the barb wire starts right on the edge of the road, but people are few and far between. Very few campgrounds, towns small and scattered, not much in the way of water. (Well, you can filter that trickle of a stream flowing through the cow pasture...if you want.) And because every square inch of land is "owned", even that trickle of a stream flowing through the cow pasture, it meant didn't see many suitable spots for wild/stealth camping. Thankfully roads are quiet, but be prepared for some long days through the Palouse to get to the next store or place to stay.

And a warning: If you use google maps, you'll notice a bike trail that runs from Spokane to roughly Pasco. "Aha!" you might say, "my route planning is done!" That's the Columbia Plateau rail-trail, which is highly undeveloped. Basically they pulled up the rails and called it good. It's just railroad ballast, doable on a fat bike, but just that. Plus there's closed trestles, and few services and water.
crazyguyonabike.com: Bicycle Touring: Columbia Plateau Trail, by Matt RC