Many people want to follow the curly girl method but do not want to give up dying their hair. Most hair dyes contain harsh ingredients that may damage hair. In addition, many contain silicones that can deposit on your hair. Luckily there are several good options:
Henna and other herbal dyes (indigo, cassia) are completely CG-friendly. They also do not do any damage to the hair. The disadvantage is it can be hard to predict the results, there is a learning curve for using them, and the color options are more limited.
That said, I have personally switched to henna and love it. I use an "auburn" mix that doesn't look too far off from my natural color. It leaves my hair shiny and healthy and I only need to dye every 4-6 months. Some people report that their curls are altered by henna, but I haven't found this to be the case.
Some CG specific henna tips
All chemical dyes contain para-phenylenediamine which damage the hair. If you chose this, you can mitigate the damage by chosing the right products for caring for your hair afterwards.
If your hair dye contains silicones and you don't want to switch away, you can clarify with a silicone-removing shampoo like Kinky Curly Come Clean after dying.
Many CG folks swear by Olaplex for restoring the hair after chemical dyes. Also para-phenylenediamine makes the hair more high porosity so you may want to switch to higher porosity products.