When I worked in restaurants, we were always looking for dishwashers. They don't stick around long, constantly pull no-call no-shows and never give notice when they quit. 

I've only worked front of the house so I don't have details, but I've seen guys get hired on the spot more times than I can count. I think they usually just bang on the back door until someone opens it and then they offer to work. 

Starting out, I'd say go through the normal entrance that customers use and ask the hostess or who ever you see to speak to the chef or someone in charge of the kitchen. 

Once you find that person, tell them you want to make some cash doing dishes. Even if they don't need you right away, they'll probably ask for your number anyways. If they ask you to fill out paperwork, just tell then you want cash only.

Keep knocking doors until you get a yes. If there's a downtown area with a bunch of restaurants, you can hop around place to place if needed. 

I suppose you can call around but I don't think they'd want to discuss cash payment over the phone. 

Best time to go/call would probably be when they're getting ready for the next shift, so around 10 am for lunch shift or 2-3 pm for dinner. 

Make sure they feed you, too. That always seemed to be included with any dishwashing job. 

Don't worry if you don't have experience, you can learn it pretty quickly and they'll train you. They're often so desperate they just want a warm body. 

Dress ready to work. Meaning jeans or some other durable pants, non-offensive tshirt, and closed toe shoes, hopefully nonslip. Sneakers or work boots are fine. You'll get soaking wet and dirty so avoid white clothing.

Good luck, you got this.