My new apartment doesn't have a dishwasher so I'm washing everything by hand. I'm leaning towards dishcloths since I can get a bunch of them and wash them once a week and hang them to dry in between uses on my oven handle.
You should wash your dishes with a sponge. Sponges absorb the water on the plate, and at the same time keep the water in so it is more efficient than using a cloth simply because the water can go out and escape, unlike a cloth, which you keep having to clean and wet and dry over and over again. Sponges also clean better and it is easier to clean because the holes allow excellent cleaning to take place for bubbles from soap.
Dishwashers are silly contraptions to my way of thinking.
Sponges carry bacteria, as the others have pointed out. You CAN wash them but ... they break down much faster than a good cotton rag. Any old rag, as long as it hasn't been used for cleaning automotive parts, works just fine.
Air dry your dishes if you can. Let leave drip dry in the dish drainer, if you have one. I use a towel on the counter to absorb the water with a glass or cup to lean them against. If there are enough dishes that are taking up too much space I use a fresh, clean drying towel to dry them off & put them away.
That's a good idea in more ways than one. Dish cloths can be washed in the washing machine and can be used over and over again. Sponges collect a lot of bacteria that we don't realize is there. I used to use sponges repeatedly until I learned this.