99% of plants grow in the top 6" of soil. A four inch raised bed is more than ample for any gardening situation at all, and is really the best.
I'll even tell you why, because that's a pretty big statement.
What you're after with a raised bed is opportunity to control soil moisture. Plants DO NOT like "wet feet". In lots of areas, the soil itself or the lay of the land does not lend itself to water control.
A raised bed allows you to improve the soil AND control the wetness of your soil. That's what they were devised for, and that's where they're indicated.
I know lots of folks just like to build them because they're cute, but lots of times they're just making a bad situation worse. They'll build a box, use the same lousy soil that's all around, put the plants up in the air, and starve them for water all season long.
If you use a plain old 4x4's, fill that sucker up with a nice improved sandy loam full of compost, and water it as little as possible, you can grow the most gorgeous plants in the world.
In case you wondered about the "other 2 inches", here's the thing...
When you use the 4" timbers, most of your plants roots will never hit the grass that is under where you started. If it does, it'll grab extra nitrogen from that thin layer of sod. If it's too much, the roots will just pull back and stay up at the 4" level.
The best part of that is that the sod will start to rot. That'll bring the worms from miles away, they LOVE to eat that stuff. Then they'll come up and worm all around in your 4"'s of dirt and improve it dramatically.
Ultimately you'll improve the soil below your frame, and that will help drainage as well.
If you want to go one more mile, get yourself a bunch of SAP, the super absorbant polymer crystal stuff. Work that in your beds and NEVER till them. Don't even turn them over. It's a waste of time and just kills your worms and wrecks your SAP. The SAP will hold water for your plants all summer long, and you'll only need to water once in a blue moon.
If you want, when you put in your transplants, you can put a little handful of composted cow poop in each plant's hole. That gives them a little kick start, and will sustain them through the year. If it so pleases, don't be afraid to use miracle grow when you water. If you do, remember the rule "weakly weekly". That means to use it at about 1/4th the recommended amount once a week.