One of my responsibilities as Greenhouse Manager at school is IPM, or integrated pest management. Used to be when you brought in soil it was completely sterilized, all the greenhouse was sterilized, then if you saw an insect it was sprayed with a chemical which instantly killed it. Nowadays, when certified organic, we use IPM. This is a combination of biological, physical and chemical controls, the main point being to create balance and long term pest management.
That said, we have a huge thrip and aphid problem, and since there are spidermites in an adjoining room, we will likely soon have those as well. I had purchased Aphideletes, a parasitic fly which lays its eggs in the aphids, but these are long term controls not short term crisis management, and we are now in crisis mode. Today I ordered lady bugs and Atheta; the Atheta will predate on the larval thrips in the soil, and will remain in the soil indefinately once colonized. The lady bugs just eat aphids, lots and lots of aphids....this is a good thing.
I also have been using trap crops. I have garlic oil, which came in a plastic jug, but has a 99/1 water/oil dilution rate, so in my mind that one jug is worth it. The oil is a repellant, so I fill a small spray bottle with the diluted oil and spray it on the leaves that I would like to keep pest free. I then leave a trap crop without the oil, for the pests to go to. I can replace this trap crop periodically and destroy the soil and plants to get rid of the eggs and larvae on the trap crops. It is not 100% effective, but it works to keep the pests from destroying the plants.
I also have been manually checking the crops and squishing the aphids and thrips by hand. While this is not something which could be done large scale, it is helping with the small amount of plants I have going right now.
Organic IPM is a measure of patience and compromise. You will not achieve a 100% kill rate right away. It takes time to work, and it will never work "perfectly". The point is, we can live without perfection. It is better for our health and the environment to have a few leaves with thrip scrape marks than to have perfect leaves full of insecticides.