How do I cure major depression without anti-depressants?

Many options. Except for severe major depression, generally, psychotherapy alone is just as effective as antidepressants alone. Though other non-pharmacological options (e.g. Light therapy, aerobic exercise, epa, etc) have been found to have efficacy, studies that support their use are generally neither as numerous, nor have as many subjects as those regarding psychotherapy.
Talk therapy or wait. Not sure why you are so opposed to antidepressants; they can be remarkably effective. However, several types of talk therapies (cbt, dbt, interpersonal therapy) have been shown to be at least equally as effective. Also, the natural course of depression is that eventually it will go away, although this may take months to years. Doing nothing usually is tempting but not a good idea. Seek help.

Related Questions

I have been diagnosed with major depression and recently I have been feeling happy with out my anti-depressants. Am I "cured"?

Discuss with your. Psychiatrist. Conditions may be in remission and may get exacerbated with life stressors.
No. May have resolved an episode of depression but if you were diagnosed with depression you probably have that diagnosis for life unless it was only situational depression. You can have periods without medications but if you recognize new symptoms then it is important to see your health care provider.