I paint primarily in oils, but I don't use turpentine or mineral spirits to thin my paint or clean my brushes. Paint thinners give me headaches. I've even tried soy-based substitute-thinner, but these too induce long-lasting migraines. Even Gamblin's solvent-free gel bothers me. One alternative is water-miscible oil paint, and indeed I have many tubes of Holbein Aqua Duo and other brands, but I prefer the consistency and choices in traditional oil paint brands. (My favorites include Winsor & Newton, M. Graham, Rembrandt, Michael Harding, and Maimieri.)
Fortunately, plain old walnut oil does most of what I want. It cleans most the paint off brushes, so it's useful during paint sessions (to clean brushes when changing colors) and afterwards (for clean-up). It also makes a perfectly nice medium for thinning paint. The only downside is that it's not ideal for laying down a ground or a thin wash before starting a painting. But one can use water-miscible oils for that, or just a thin couch of walnut oil. I'm still experimenting. If anyone has thoughts on how to make a solvent-free underpainting, I'd love to hear them.