Piece o' cake. From 30th and Belmont, head south to Lincoln (four blocks south of Hawthorne). Either 30th or 34th works well for this leg of the trip. Turn left, follow Lincoln all the way to the Mt. Tabor Park entrance just after you cross 60th. Veer left to enter the park and follow the road up until you get to the first Y. Stay to the right and head back downhill.
You'll leave the park on Harrison St, stopping at 72nd. Continue straight until you get to 76th. Turn left, then take a quick right onto Stephens. Go straight until you get to 80th, turn left, then take a right at Mill.
There's a signal at Mill and 82nd for safe crossing. Go over to the sidewalk and hit the ped crossing button, because the loop detectors don't seem to find my bike. Otherwise, you'll sit there until a car comes along to trip the light.
Continue straight on Mill until you get to 92nd. Cross 92nd and go straight until you hit the I-205 bike path. Take a left and go one block until you get to Market. You can either turn right on Market and take that up to the hospital, or cross Market and go to the ped crossing you spoke of. But there's really no need to use the ped bridge; Market is a low-traffic street and you won't have to worry about climbing the grade up the ped bridge.
The climb on Mt. Tabor is pretty mild and short; You're really going around the mtn, rather than over it. If you're absolutely against any hills at all, taking the Clinton St. bike route is the next best option. It's further south than the Lincoln St route, but it avoids Mt. Tabor. The downside is that it's longer and a bit more convoluted once you cross 82nd Ave. Plus you have to ride a bit on heavier trafficked streets.
Good luck...and I hope none of us ever meet you while you're on the job!!! :/