I say, Don't believe the hype! There are many great little pocket neighborhoods in Oakland. You might want to check out Maxwell Park or Dimond Heights. Drive through Laurel and down into the Allendale Park areas. You will find many happy families flourishing, I guarantee it!
Best of luck and I hope to see you around the neighborhood!
First, the neighborhoods that within the Oakland city limits. Piedmont, is excellent. It is in the hills of Oakland good schools and parks for kids.
The next would be Montclair. This is a very nice neighborhood very close to Joaquin Miller Park witch is very scenic and a great place for walks and places for kids to ride their bikes. It also has lots of little local shops and businesses that have a nice charm to them.
Last but definitely not least, is Alameda. This is the town that I grew up in. Technically not in Oakland's city limits. It is a city on it own. It is a small island right on the outskirts of Oakland (100 feet of water separate the two) It is a safe quiet little town that has everything that a family needs. Good schools, nice homes, local shops and businesses. It also has an excellent park and rec dept that has everything from little league to after school programs.
These three locations plus a few more should give you a place to start looking in the East Bay.
Hope this helps