There hasn't been a time in my adult life during which I haven't done dishes by hand. That wasn't always by choice. Most of the time, I dreaded it. It's been only in the last few years that I embraced doing the dishes as something that I actually enjoy doing. I think the shift comes from a change in mindset, which for me includes a positive attitude, the conscious experience of the activity, and sometimes in addition an intentional practice of gratitude. So here's a short guide on how to relax while doing your dishes.
The first part, your attitude, is simple. Switch off the inner dialogue that says, "Ugh, I have to do the whole freaking pile of dishes, again". You don't need to turn the ship around completely and brainwash yourself with happy thoughts, just stop the inner whining.
Secondly, fully "experiencing the activity" refers to your capacity to be in the moment instead of getting lost in either planning, fantasizing, or recounting old stories that happened at a different time or place than you are in now. Instead of letting the mind wander, stay with what you are doing and what is happening as you hear it, see it, feel it, and smell it. For example, consider how the water feels. Is it warm, is it light, is it getting oily? What noise does the brush make when you scrub the pan? What background sounds do you hear? Any birds, barks, traffic, wind, music? How does the soap smell? How is your body feeling while you are standing bent over the sink? Where is it loose, where does it feel tight? Explore your environment without sorting what you find into good/bad, like/dislike categories. This practice, to bring your mind back to where you are without judgement, is in itself calming and relaxing.
In addition, if you like gratitude practices, you can recall the food that was on each plate as you wash it. Consider the ingredients that found their way into your home and nourished you and the people you live with. Think of what filled your cups and glasses. For example, imagine the journey of a tea leaf from the elements that allowed it to grow, the farmer who picked it, the workers who dried it and packed it, the truck drivers, dock workers and seamen who transported it, the grocer who stocked it, your hands which measured the tea, steeped it in hot water and poured it, and the friend who shared a conversation with you as you drank the tea together. Send a mental thank you to everything and everyone involved in the making of that cup of tea which you are cleaning in that moment.
You can also practice gratitude by appreciating all the things you are using to do the dishes. How amazing is it that you can turn on the faucet and clean water is running? How cool that you have hands to wash with? How fabulous that you own cups and plates that serve you when you need to eat? How exciting that all your senses work to let you experience this totally awesome moment?
Aside from attention and gratitude practices, doing dishes can also be an opportunity to take stock where you are in your day, to pause and reset, and to move on with a clean plate, a clean slate, and an open mind.