What size do the cushions need to be? Joann Fabrics has 24X24X3 premade pillow forms that were a good size for the chair cushions I just replaced. They are polyfill, a little pricey but sign up for coupons and get them for 40% off. I used a polyester upholstery fabric just because it was the only patterned fabric in the store that I liked for my particular purpose. It smelled funny for a while even after I washed it. Any heavier weight fabric should be fine. You can use cotton. Just prewash so it doesn't shrink after you sew it. I made the cushion covers with zippers so I can wash them.
Polyfill pillows will probably flatten with use. Pillows with a foam core will spring back better over time. You can find greener foam options in standard sizes. I think they are made with corn. People that use foam usually wrap it with a layer of batting. The batting layer makes a big difference in comfort when you use foam. You can buy cotton or polyester batting. If you do that, I would cover it all in a lightweight fabric (muslin or an old sheet) so it all stays together if you want to remove and wash the main upholstery fabric.
The cushions I just replaced were really old cushions that actually had springs inside. The springs were covered in muslin. Then they had a layer of straw-like fiber, followed by thick cotton batting.