I'm with the mnemonic / memory check camp, if you're flying simple aircraft or are familiar with the type. That's 2 quite big "ifs"... if you normally fly a plane with a carburettor engine you'll wish you'd used the checklist when trying to start an identical looking fuel injected model!
I was taught to fly without using checklists, using a "back to front" then "left to right" cockpit check pre-start, and mnemonics for pre-take off and circuit checks. Emergency checks had to be memorised.
Post PPL on a new type, I got into the habit of using a checklist. One day I skipped a line by accident, and almost took off without locking the canopy - a potentially fatal mistake in a Slingsby T67. A pre-take off mnemonic check would have been safer.
Someone above said that checklists are safer when used properly. On multi-crew aircraft, one person reads from the list while the other makes the check. Nothing gets missed. When you're alone, it's too easy to get distracted and miss a line...
In my view, everyone should use a quick memory check of the things that could kill you on take off & landing.