There are a number of issues with this. One, Los Angeles is VERY sprawling; people are traveling from every direction to every other direction, there are no easy routes you can make that will cut very much traffic. You would need to turn a LOT of streets into one-ways with bike lanes.
Two, the traffic density and management is crazy here. So many traffic lights and stop and go traffic. Major streets are constantly crossing other major streets, which require long stops at lights while the other major street moves through. Riding a bike in the area is especially hard, because you are constantly having to come to a complete stop, wait for 3 minutes, and start again.
This. It's also really, really annoying that all the lights on an entire stretch of a street cycle at the same time. It really fucks up traffic. To make it even worse, people often try and race to the other side even on yellows and block up the intersection, which makes traffic even worse.
I didn't realize it was such a huge problem until I moved to Los Angeles a few weeks ago. We did a bang up job finding an apartment with grocery stores, restaurants and parks within walking distance. Unfortunately, the red light situation makes it somewhat difficult to get to these amenities.
LA needs to do a better job of balancing pedestrian and motorist interests. With more and more people abandoning or at least de-emphasizing cars, I hope and expect this balancing act to happen sooner rather than later.
I see it happening in different parts of LA. Santa Monica and West Hollywood are starting to adopt the "rent-a-bike" system that I've seen in NYC and Denver. They're also starting to set up more bike parking areas to be more bike friendly. I don't know how well it will work out in the long-run, but I do really hope to see more bike lanes in LA to encourage bike commuting.