What could I do to soothe three herniated disks in my spine, other than surgery or medication?

Options. Typically, only one of the discs is the main source of pain even if the MRI shows multiple. The majority get better on their own, but antiinflammatory medications taken for 1-2 weeks at a time can help. I also often recommend traction with a pt or chiropractor or by gravity when using an inversion table. An epidural injection can also help the nerve irritation caused by the herniations.
Several things . Time is the biggest healer as 90% resolve with no surgery. Lifestyle changes like weight control, not smoking & regular exercise are important. Core strengthening & cardiovascular exercises are important as well as flexibility. This can be learned through physical therapy. Some people like to use inversion tables for traction or try acupuncture or even chiropractic .