There's not a *great* way to do it, but you can.

The Interurban trail runs through Fort Dent up past the soccer fields. Continue on the trail as it goes up paralleling and then over the river. At the south end of the Foster Golf Links, turn right and go up on the sidewalk. Continue on that until you get to I-5. If you look, you'll see the trail taking off to the right just before the on-ramp (PAST the bus stop and the piano store). Follow this and you'll keep following the river. At one point, you'll pass under Highway 99 and make a circle turn to go over a bridge over the river to the trail on the other side. Continue on the trail, going over the river again, until the thing basically ends. This will be at an intersection where there's an off-ramp from West Marginal Way. Turn left there onto the frontage road and take that up to 14th street (it's a straight shot to get onto 14th, although I suppose technically it's a right turn) into South Park. Continue on 14th across the South Park drawbridge up to East Marginal, where it T intersects at Boeing Field. Turn left onto West Marginal and continue up to the light at Ellis. Turn right on Ellis and continue to Airport Way. Turn left on Airport and continue through Georgetown, over the railroad switching yard, through SODO, and all the way up to Royal Brougham. Turn left onto Royal Brougham and follow that all the way down to Alaskan Way. Turn right on Alaskan and follow it all the way up to the top of the waterfront where you'll find Myrtle Edwards Park. From there you should be able to get to the BG pretty easily.

Note that once you hit the intersection where the West Marginal off-ramp is, you'll be riding on streets, not trail. It's not usually a huge problem, because the streets are generally wide enough for a bike and a car, or they're multiple lanes in each direction, which means you can take the lane without having to worry too much. The railroad bridge is a bit of a bear to climb, but it's relatively short. I ride the part through South Park up to the waterfront every day and haven't had a problem except the occasional buzzing car or honker.

Once you've done it, you'll have no problem figuring it out. I've done it lots of times, and it's actually a rather pleasant ride. Fairly flat, and it parallels the river quite a bit.

King County bike maps can be found here http://www.metrokc.gov/kcdot/roads/bike/map.cfm if you want to look more closely.