Not asking for much, are ya? ; )
1. Get on Auburn Way N.
2. Auburn Way N. becomes Central Avenue around "Thomas" and into Kent.
3. Make a right turn on Kent-Kangley Road (WA Hwy 516).
4. Make a left on 104th Avenue (WA Hwy 515), which eventually turns into 108th Avenue SE.
5. Stay on WA515 under/over I-405 (in Renton), past the courthouse, until you reach a junction for WA Hwy 900/169. That may be NE 3rd Street. If so, turn right. If not, turn right when you see the Historical Society Museum.
6. Stay on NE 3rd Street - it becomes NE 4th Street.
7. Turn left at 138th Avenue SE - that may become 95th Way.
8. Turn/jog right onto Coal Creek Parkway SE - this runs up through Newcastle and by a golf course before hitting Bellevue.
Why exactly did you want to avoid the freeway? It would have to be faster to take WA Hwy 167 to I-405.....unless you don't like high-speed driving.