It's a crap shoot. There will probably be a historical bias for one direction, but in any given year, anything is possible. Also, the historical data will be different for different months, so it depends on when you're going.
It also varies along the route, but suppose I pick Spokane (about midway) as a representative sample, and let me pick the month of July at random. The bias for wind direction for July in Spokane is from the southwest. That suggests you might get some slight benefit of going west to east, but the benefit is just a probability and will likely be small. You will still get some headwinds, and those crosswinds will feel like headwinds anyway.
Note that if I were to pick January instead of July, the bias is exactly opposite. There are slightly stronger winds from the northeast in January, although winds from the southwest are in very close second place.
In April, the southwest bias is even stronger than it is in July. In October, the winds come from almost every direction except the northwest, with southwest and northeast winds still occurring more often than other directions, although there are also a lot of winds from the south.
Winter is significantly windier in Spokane than summer, so choosing the season is more of a factor than choosing the direction. I doubt you'd want to go in winter anyway, and July and August are the least windy, so that's probably when you want to go.
Bottom line: Use some other criteria to decide which direction to go.