I would look to do this if I was diagnosed with Hashimoto's thyroiditis tomorrow.
1) Try natural thyroid medication.
2) Check whether I was low in Selenium and iodine. If low, and safe to supplement. Do a week on 200mcg of Selenium, and then start on low iodine or kelp tablets. Iodine overdoses are possible, and can lead to detox. Contact an integrated practioner. For similar effects, and safer, just supplement selenium. Drink soda water regularly when taking kelp/ iodine to deal with the effects of detox.
3) It's critical to give up gluten. This I would recommended to everyone as it's safe and research indicates it's necessary to feel better. I would talk to a nutritionist for help adjusting to a gluten free diet. Your GP can refer you to one.
3) Talk to a doctor about being tested for vitamin d, iron and b12. These nutrients are critical to optimal thyroid patient health in my experience.
4) Accept I have this disease and will likely need medication for life to feel human. Denial and skipping medication will see me five to twelve unnecessary kilograms heavier, weak bloated and miserable.
5) Learn to listen to my body and nurture it.
6) Learn the signs for being over or under-medicated and talk to a doctor for advice and testing as necessary.
7) Find a M.D. I trust for moral support and advice.
Disclaimer: this is the writer's opinion only. For medical advice and treatment speak to a medical professional.