I just had a bunch of huge trees removed, the tree service use a machine that grinds the roots below the ground, it took 2 mins per tree. I would suggest going to a neighbor or "servicemagic.com" and checking the reviews from previews customers, select a tree removal company and ask them to come and grind out the roots. That would probably get rid of most of them and leave some nice soil. I reckon its worth the money otherwise you would be there all summer breaking your back trying to dig them up!

kevingalaxy, I went on serviemagic.com. They have stump romoval service listed but not removal of roots under the ground.
How can I find people who can remove tree roots under the ground on servicemagic.com?
I don't need to remove stump but only need to get rid of many roots running around the stump under the gound.
Please excuse me, if I sound stupid because English is my second language.

Ever think of building some raised beds? Then you don't have to worry about the roots.

bogey123, yes, I thought about making raised bed but because the area I want to expand is large and price of material for raised bed cost over $2000.
It is cheaper to hire tree service people if I can trust them.

Sakura,
It will cost you more than 2000.00 to have every single root from a tree dug out of your yard, and that is if you can find someone to do it.(good luck)
Your entire yard will be literally torn up.
You can't do that.
The roots will eventually rot in the ground.
It will only take a couple of years, or longer, depending on what kind of tree it was and weather conditions.
Buy cinder blocks in the meantime and fill with potting soil.
You can't get any cheaper or stronger than that.
Research on this garden web the words "cinderblock" and raised beds. Many people do this they work great!
It is what it is.

Sakura,

Not sure what something like that would cost. But if you are in the Delaware/Chester Counties area of Pa, I know someone who does that kind of work.

You could call him and he'd come out and give you a free estimate. Heck, I think most tree service folks would be happy for the work these days. I'm sure you could call and get several to come out to give you free estimates.

Honestly, it doesn't seem like a lot of work to me, if you have the right tools. (And a shovel and ax wouldn't be the right tools)

It sounds like you need to rent a rototiller.
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A rototiller would do nothing for removing tree roots. Hire a backhoe service to dig it up and remove the roots if they are that bad.

raised beds are your most economical solution.

Backhoe isn't the right tool...unless they are really large roots. Ripper tines on a tractor will do it for roots up to a couple inches in diameter, but that process will take out the entire yard. Usually landscaper blades (box blade) comes with rippers. If you really want them out, then that's the way to do it. Consider it the continuing price of taking out that large tree.