Giant Tree Roots All Through A Yard
 

I have heard of people putting wood chips around their trees or putting new fill dirt and grass, so that it can hide or places them back underground where they should be.

Tree Root "Network"
 

Aside from making yard work difficult (and being a tripping hazard), my concern is that - even farther underneath than where they're obvious - they may be "planning" to create problems around of under the foundation of the house.

The other thing I imagine is if we were to somehow cut through them ourselves whether or not we'd end up killing whatever trees they're attached to.

I know we could call "a professional", but this particular job is something I'd rather be able to do something about myself.

This seems to be a pretty hard job to do for yourself. I suggest that you at least call a professional to see what options they have or what they suggest. Of course, they'll tell you that your garden is in danger so you hire them but just do some research around to see what your options are.
This sort of job needs to be done correctly from the beginning so it doesn't become a pain later on.

Good luck!

That's a more immediate concern if the trees are right next to your house. 40 year olds usually have a massive root system and what you see at the surface might just be the tip of the iceberg. A pro can identify your trees and determine their rooting habit. It might be necessary to girdle and kill them. :(

I had an instance of this where the roots of my trees were causing cracks in a free standing wall. The amount of work to get all the roots removed was unbelievable.

On the other hand as well I have had a friend that cut down all his trees in the garden, and it ended up almost like a swamp, because of the amount of water the trees absorb....

That could be an issue. An arborist or forester could usually determine the best options considering all factors involved (species, climate, soil, and water). Sometimes the simplest route isn't the best to take.

Yes, there might be a better solution compared to what you might think is the easist one..

Sometimes it is better to get the professionals in to give advice.

Invaluable advice. This is exactly what I have done. It's also rather attractive!