Is there such a thing, you may ask? Yes, and you are the one to make it. You can actually perform composting on the materials found on your garden. You can use fallen leaves and twigs, animal manure, but this will depend on the type, and many more.
This type of gardening wants, as much as possible, to stay away from pesticides. But if you can't afford to pick the insects one by one by your bare hands, then you can go to your local grocery store and ask for an organic pesticide that is available commercially.
The old way really is to be vigilant with your garden and take off every pest that you see. You should only turn to the organic pesticides when it becomes too many and uncontrollable that you can no longer handle. You can also try to bring in the animals that feed on those pests. This way, you'll have some help in picking those pests up. And that is also helping the other animals satisfy their hunger.
As a hobby, this may be time consuming. So if you cannot devote enough time into it, might as well find a partner or drop the idea until you've found the right time to carry on with the tasks.
Organic gardening really entails a lot of hard work. So you better be prepared to perspire in the process. To ease your tiredness when you are already into it too deep, just think that what you are doing is helping nature. This is your way of giving back what nature has bestowed on you since the day you were born.