Art (well named, isn't he?) at M. Graham and Denis at the Art Shack shared with me how to use oil paints without solvents. Above are baby food jars that I fill with cooking oil to rinse out my brushes. Swirl your brush around and then press it against the rim to let it drip. You may also want to put a bit of plastic mesh in the bottom of the jar to rub your brush against. This helps even more in cleaning out the paint. To avoid cross contamination of colors, use a different bottle for each color. When my jar is full, I empty it into a container suitable for hazardous waste (like an empty house paint can). From what I understand, it's best that pigments of any kind don't go down the drain and get into our water.
Depending on how thick I want the paint, I can apply it straight out of the tube or thin it. Linseed or walnut oil can be used depending on your preference. Walnut oil does take a bit longer to dry, but is less yellowing. That's nice for the lighter colors, especially. M. Graham also has Walnut/Alkyd Medium that speeds the drying time. If you're thinking of trying it out, here's an info page on their stuff. It will take about five days for your layer to dry, but this can have its advantages results wise.
The Art Tree House , a site that offers safer art supplies, has another alternative. Spike oil of Lavender is a type of solvent that is less toxic than some of the more common solvents. Do you like the smell of lavender? Well, imagine that scent throughout your studio. Sounds nicer than turpes, doesn't it? Personally, I'm sensitive to lavender so I know I can't use it, but it may be useful to you. Only a drop or two is needed. Solvents are always stronger than the oil method, so if you're sensitive, I'd recommend the oil method.
I've found that I react to some colors and not others so that would be a problem with the pigments, perhaps certain metals. I may end up doing chelation to get rid of heavy metal toxicity and since that's related to the paint sensitivity, will let you know if it works out.
If you have any questions, don't hesitate to ask in the comments or by my email address (listed on my profile page).