Check out your local craigslist. I get most of my wood from there. Usually I will trade a bowl for some logs of a tree that someone has cut down.
The wood is usually wet (freshly cut) but then I just cut the logs in half and seal the ends.
If I want fairly round bowls, I will rough turn the logs into bowl blanks (about 1" thick), seal the end grain and then toss them on the shelf for 4-6 months. Then I will re-turn them into finished bowls. They usually stay pretty round this way.
Most of the time, I will turn from green wood to finished bowl. The green wood is easy to turn and once done will distort some. I think the distortion gives the bowls a more natural look but my wife prefers the twice turned (round) bowls.
Another place to look would be a cabinet shop. Sometimes they will have 8/4 and 12/8 scraps that you can buy for cheap...