I've found two things make a difference - your dishwasher, and the detergent. You definitely get what you pay for with detergent. I have just stuck with Cascade, which seems to work best.
Your dishwasher will make a huge difference. The dishwasher at our old house (lower end model, about 10 years old) is just like you described. If I didn't rinse off everything, there would be little pieces of crud on the dishes. Yuck. My new dishwasher is completely different, although I still do rinse some dishes (old habits die hard). :rolleyes: