I traveled HWY 14 east of Portland a number of times last summer and went through the series of tunnels just west of Hood River bridge. Choo Choo Man 74's pictures show exactly what it looks like. The posted signs show the clearance at the solid white line on the edge of the road. During my trips I saw at least half a dozen large semis going through the tunnels with no problem. The tunnels are fairly short and traffic seemed to be very light. Most of times the semis eased towards the center of road as they went through, but it looked like you could safely ease back to the right if someone was coming the other way. Your call on that.
Not sure where you are headed but if you are headed east on HWY 14 from Portland you could drop down across the Columbia River at Bridge of the Gods (near Stevenson) to Interstate I-84 and head east from there. It is a short drive from there to Hood River Bridge and then on to The Dalles. I am pretty sure you can cross back over at the Dalles and avoid the whole tunnel situation. Check it out. Good Luck!