Neem will work on quite a few bugs (depending upon how hungry they are). Surround will also keep a lot of bugs off your plants, but does require reapplication after rain and makes your plants white. There are several organic insecticides (pyrethrum based ones, and others) that are quite effective, but not that great for bees.
If you could identify who is doing the damage, then you could more easily pick the most effective one for the bug that is causing your problems. Otherwise you are going for broad spectrum solutions which may or may not be as effective, and may end up being more potent than you needed.