I keep an organic garden and have found a combination of biological control (nematodes for cut worms and beetle larvae), crop rotation and the occasional spray of Surround (a fine Kaolin clay sprayed on hard fruit, keeps the bugs off), neem oil (dormant spray for my fruit trees and canes) or Serenade (B. subtilis to prevent various forms of mold including blight on tomatoes and powdery mildew on soft fruit) does the job quite nicely. None of these are an issue for honey bees. A spray of dilute soap solution takes care of the occasional outbreak of aphids though I've not had one of those in the last 3 years. If you have chickens or guinea fowl I highly recommend letting them loose in the vegetable garden as long as you don't have any small seedlings. They seem to be very effective at mopping up anything bigger than an aphid that's above ground but leave the bees alone.
Hope this helps,