Unless you are looking only for restaurants you can walk to, you are in an ideal foodie location. Many of LA's best restaurants are within a couple miles of you: Bazaar, Lucques, Campanile, basically everything on Third Street and Beverly and La Brea. You also have great good places like Ed's Coffee Shop on Robertson, Farmer's Market (DOUGHNUTS!!). This is a ridiculously abbreviated list of what's close by, but you will discover that by taking a couple of neighborhood shortcuts, you can get to enough restaurants within a 15 minute drive to keep you busy for a few months. Not to say that there aren't other great restaurants in other parts of town--but your area of town is more of a foodie oasis than a foodie desert.
8474 Melrose Ave, Los Angeles, CA 90069
Ed's Coffee Shop
460 N Robertson Blvd, West Hollywood, CA 90048
You MUST try Bloom Cafe on Pico, maybe a mile east of La Cienega. It has awesome food and juices, and has tons of healthier types of food like mescaline and cous cous. Great for brunch, too. We moved to the area a couple months ago and quite frankly this was our saving grace (without driving West into traffic or North into traffic...
Cafe Bella Roma on Robertson has good branzino and other fish and wonderful pasta. bloom Cafe on pico near Fairfax is worthy. La Provence on Olympic is good for breakfast, lunch or pastries. Matsuhisa isn't far nor Joan's on Third.
129 N La Cienega Blvd, Beverly Hills, CA 90211
Cafe Bella Roma
1513 S Robertson Blvd, Los Angeles, CA 90035
Check these prior threads out for potential ideas: