Basically you can use anything you want. Sheets are not woven any differently than off the bolt cotton fabric. They both have threads going vertically and horizontally. That's called the grain of the fabric. All cotton fabric has that no matter what industry turns it into. It's what you do with it that's important.
I've used new sheets, old sheets, and fabric off the bolt. A lot of it will depend on if you are planning on hand quilting or machine quilting. If you're hand quilting, old sheets or off the bolt may be easier to needle for you. New sheets still have a lot of sizing in them and it takes quite a bit of washing to really soften them up for needling. Off the bolt may have a lot of sizing also. As you get used to touching fabric, you will begin to learn the feel of fabric with sizing.
You will also need to consider the type of batting you're going to use as well. I usually use polyester for hand quilting. I find it needles better than cotton or the cotton/poly batting. For machine quilting, I have found cotton works best. Each type of batting will create a different effect.
It's completely up to you on what you want to do. Keep in mind that everyone has different experiences and has had different results. There really isn't a 'right' way or 'wrong' way. There's only your way. If something doesn't turn out quite the way you thought it would, just stand back and say 'I meant for that to happen' and continue on.
I look forward to seeing your finished quilt.