Here's a link on the Cascade website that leads you to a number of area bike maps. I'd suggest you take the Bellevue map w/you when you are house or apartment hunting.
http://www.cascade.org/Community/maps_routes.cfm
As you can see, a couple of the routes across Bellevue are marked "caution", they're pretty busy -- but not impossible. A bunch of other routes are just fine.
Also, you may want to consider living up in Kirkland and biking down to your job, or even up in Redmond, and cycling across on the north side of 520.