I've met friends with their pooches at Amandine a number of times, and it's always worked out very well.
Further East, Cafe Midi on La Brea always seems to have a bunch of dogs and their people at the tables outside. I think Midi will even bring water bowls for the dogs while they wait for their people. :)
148 S La Brea Ave, Los Angeles, CA 90036
lots of places on Montana like Blue Plate, Babalu, Kreation Kafe, Stefan's, Cafe Montana
Santa Monica - Bread and Porridge, Bar Pinxto, Locando del Lago, South, Novel Cafe, Snug Harbor, Huckleberry
Beverly Hills -- Urth Caffe, Beverliz Cafe, Greenleaf chop shop, Panini Cafe, Joan's, King's Road, Bossa Nova, Euro Caffe, Coupa Cafe, Cafe Fiore, Il Pastaio, Ingrid's, Frida, Xi'an, Cabbage Patch, Enoteca Drago, La Conversation
267 S Beverly Dr, Beverly Hills, CA 90212
638 N Doheny Dr, West Hollywood, CA 90069
1002 Montana Ave, Santa Monica, CA 90403
Kings Road Cafe
8361 Beverly Blvd, Los Angeles, CA 90048
Il Pastaio Restaurant
400 N Canon Dr, Beverly Hills, CA 90210
2323 Wilshire Blvd, Santa Monica, CA 90403
419 N Canon Dr, Beverly Hills, CA 90210
410 N. Canon Dr., Beverly Hills, CA 90210
308 S Beverly Dr, Beverly Hills, CA 90212
1415 Montana Ave, Santa Monica, CA 90403
214 S Beverly Dr, Beverly Hills, CA
1534 Montana Ave, Santa Monica, CA 90403
66 California Street, Ventura, CA 93001
9601 Santa Monica Blvd, Beverly Hills, CA 90210
To the east of LA, I was allowed to bring my dog to sit on the Patio of the Back Abby Tavern in Claremont. I think I just had a drink that time and not too many people around.
Just outside LA county, Molly's Super in Upland catycorner from the Public Library (and the city admin offices) is super dog friendly. There is a section of the patio reserved for dogs, with water bowls, free treats, a lead staked to a tree and you may sit right next to it. There is also a dog menu if you'd like to order your pooch a meal. The friendly owner is such a dog person, she treated me to lunch the time I went there with my new pet from the shelter. It was a thanks you for rescuing the doggie. Closes at 2:30.
We drive up PCH to Malibu with the dogs all the time. Lots of places to take out at the Malibu Country Mart and you eat at the tables outside. Or Malibu Seafood. They don't let you eat at their tables, but just next door are tables that are part of the parkland. Also, outdoor tables at Neptune's Net on the right side of the lot.
I know this is an often asked question but I'd vote 3 Square on Abbot Kinney. Great food but somewhat cramped patio area. They don't mind you sitting on the outer edge of the patio with your pup just on the other side of the railing. There might also be some tables outside the patio bordering the sidewalk which would allow you to sit directly with Rover. Bring his water bowl if going for lunch - the sun will be cutting directly across the front of the restaurant at that time.
3 Square Cafe
1121 Abbot Kinney Blvd, Venice, CA 90291
Just about any part of town that has some history of dog culture will have places that you're looking for. Venice has been a "Dog Town" since I was a kid, so it's pretty much a given around here. As long as there's some sort of outdoor/patio area, particularly if it's adjacent to a sidewalk or parking lot, then most places will allow dogs to be on the perimeter.
In Santa Monica we see lots of well mannered dogs on the patio at il Forno on Ocean Park Blvd. (and they have some very good food). The patio tables are spread out, far from the street (so they are quiet) and the ones along the window are actually separated by tall planters with heavy bushes growing between them, so you have real privacy for you and your companions (2 and 4 legged variety).
Il Forno Caffe & Pizzeria
2901 Ocean Park Blvd, Santa Monica, CA 90405