I live in LA and just started commuting by bicycle a few weeks ago as well. My route begins about exactly where yours ends - I live in West Hollywood and work at UCLA, about 6-7 miles each way.

Between my neighborhood and Downtown it seems like you have a lot of options for east-west travel, as there are lots of straight parallel streets. Have you ever tried using Fountain? It runs parallel to Melrose and Santa Monica, and is between Santa Monica and Sunset. It is a designated bike route for part of the way (though there is considerable traffic during rush hour - still, it isn't Wilshire Blvd). Also, in the western part of West Hollywood and towards the City of Santa Monica, Santa Monica blvd has a real bike lane for some of the ride.

I am intimidated by the traffic too, but have been riding on the major streets anyway. There is one stretch of Wilshire near the country club that is so bad I have to use the sidewalk, but the rest of the time I use the streets.

Riding in LA can be difficult, but there are a lot of resources (and the weather couldn't be much better). Have you checked out the Bicycle Kitchen in Hollywood?

Also, if you are into night riding and exploring interesting points of LA history/geography, you might want to check out RIDE-Arc (www.ridearc.org).

Best of luck-