My apartment doesn't have a dishwasher, so I wash all of mine by hand. I rinse out dirty dishes (complete with non-soapy sponge) as soon as I'm done using them - that way I don't have to worry about them getting nasty if they sit on my counter for a few days. lol And it keeps the water clean when I do wash them.
I fully intend to have a dishwasher when I have a house, though. And for that, food would be scraped off into the trash, but otherwise it goes straight to the dishwasher. That is what I did growing up and our dishwasher did a great job (and with a family of seven where all meals were homecooked, it saw a lot of action!)
ETA: our dishwasher was a power house (not cheap), and mom would soak really groady dishes in the sink to help loosen stuff up before placing it in.