The first thing you will want to do is be sure you are actually having a cutworm problem.  Cutworm damage is above ground on leaves and as in your case they cut young plants off at the soil line.  However,  slugs also do the same thing. Cutworms will leave a clean cut where they chew through,  slugs dine from the side of the plant and leave a ragged edge.  Unfortunately both cutworm and slugs are pests throughout the year.  Several moths with different egg laying cycles produce cutworms.  Waiting for them to mature and move away can be a losing proposition.

The best way to get a positive identification is to examine your beds after dark with your flashlight.  If the larva are found, remove them by hand and destroy.   Nature will be more than willing to help.  Birds, especially ground scratching birds,  ground beetles and spiders are the cutworms natural enemies.  Understanding that you want to be mostly organic; pyrethrins and spinosad A&D both come in formulations that are OMRI-listed for organic use.  Follow the directions on the label to insure maximum damage to the cutworms and greatest amount of safety to yourself and your garden.  In the meantime remove weeds,  grasses and other debris that provide cover for the feeding cutworms.

If it turns out that slugs are your problem, Oregon State University entomologist Glenn Fisher recommends Iron Phosphate for slug and snail baits.  Unlike some other baits they are relatively non-toxic to children and pets.   Once again follow the instructions on the package to achieve the best results.  It is important to remember that baits only kill the slugs that eat the bait so eradication is slow and never complete.