Unless you really want to see Missoula - which is a great town BTW - you can take Wash 14 to I-82 - cross the Columbia River at Umatilla - then continue along the south bank via US 730 and US 12. There's a nice, quiet back road through Juniper Dunes to Washtucna. Then you head up to Ritzville and take back roads through Sprague to Cheney.
Two options from Cheney -
A) Via the incredible, paved Trail of the Coeur d'Alenes from Plummer, ID to Enaville and then over Thompson Pass to Thompson Falls. From Thompson Falls you can either follow the Thompson River on dirt to US 2 or follow Hwy 200, Hwy 28, and US 93 to Kalispell and up to the NT.
B) Via Spokane and the Centennial Trail into Idaho - then up back roads west of US 95 to Sandpoint and the NT.