All of the following products offer a natural alternative to popular chemicals.
Cinnamite - natural cinnamon based miticide, kills many others insects also; works great on tender, young plants and larger plants; short term killer
Insecticidal Soap - natural extract made from fatty acids of potassium salts; works great on tender, young plants and larger plants; economical, short term killer
Neem Oil - natural extract from the neem tree, used for ages, now “redisovered”; works great on tender young plants and larger plants, degenerative killer
Nematodes - natural, parasitic organisms which attach to the larval stages of insects in the medium; provides control on a variety of insects; degenerative killer
Pyrethrin - natural extract from chrysanthemums; works well on larger plants or room infestations, look for products without piperonyl butoxide; near instant killer.
Rotenone - natural insecticidal extract; works well on larger plants or room infestations, look for products which contain pyrethrins also; near instant killer.
Yellow sticky traps - “natural” insect trap; works well as an indicator of potential problems and does an excellent job of controlling flying insects; slow, tortuous killer 
In the fall, ladybugs are all over the window sills and are easy to bring in the garden to eat some pests and keep the plants company. They rarely seem to get stuck in the resin. I didn’t list predators because they haven’t worked well for me and won’t survive an application of pesticide any better than the pests. I listed nematodes because their application is similar, but the predatory insects available are options when trying to avoid chemical pesticides. Something important to organic advocates.
Natural insect repellents
Garlic and hot pepper extract sprays are useful to keep bugs off of plants. Garlic Barrier and Wilder’s Hot Pepper Wax are commercial offerings of natural insect repellents. I use the Hot Pepper Wax on cuttings to help them retain moisture and keep insects from populating under the dome.