That's a tough one. I suppose some folks would tell you not to remind them, and not to wash their dishes. When there are no clean dishes for their meals, they'll start washing them. But that to me just seems harsh and not the kind of home I'd like to live in. It's not the way I would want my husband or someone else to deal with me, if I didn't handle some household responsibility.
So what CAN we do when kids don't step up to the plate and do their "chores" without being reminded? It certainly can be a pain to have to remind them, and it's easy to get resentful and irritable about it.
I wish I had a magic wand that would motivate kids to clear their dishes and put them in the dishwasher. The closest thing I know to such a wand is your kids' love for you. Meaning, when your kids know that doing something will consistently get them a smile, hug, or warm thank you, they're more likely to do it. By contrast, if we think they should do it without reminders, and the whole interaction is fraught with irritation on our part, they're more likely to shy away from doing it. Sad, but true!
So my advice would be to keep reminding them, every single meal, and help them create a habit. After all, they don't have a lot of incentive to put their plates in the dishwasher, so the only reason to do it is that you'll be in their face (in a nice way) reminding them until they do it. After awhile, it will simply be a habit, and most of the time you won't have to remind them.
I hear that you feel you're nagging, and that you're tired of hearing yourself. But reminding doesn't have to be nagging. I think which category your reminders fall into might depend on your tone of voice. Is there a way you can get playful with this? Experiment with being silly and playful and ridiculous about it, until everyone is laughing about the dishes getting cleared. The anxiety will disappear, and any power struggle will disappear. In fact, your disappointment about having to remind your kids will disappear. And once there's lightness and fun about it, you might even find that your kids no longer need prompting.
Good luck, and please let me know how this goes!