Is Woodford's "cashless" economy a model of a monetary exchange economy, or a model of a barter economy? (In a monetary exchange economy, all other goods are only exchanged for one good called "money"; in a barter economy each good can be exchanged for any other good.)
If you answer "barter economy", then modern New Keynesian macroeconomics looks like Real Business Cycle theory with sticky prices.
If you answer "monetary exchange economy", then modern New Keynesian macroeconomics looks very different from Real Business Cycle theory, and looks like a continuation of the Keynesian/Monetarist research programme.
Sorry to keep repeating myself, but I think this question is important, and underlies a lot of confusion.
(I think it only makes sense as a model of a monetary exchange economy. Otherwise underemployed producers would barter their way back to something approximating "full employment" even if the central bank set the nominal rate so that the real interest rate was stupidly high. "I will buy more of your overpriced apples if you agree to buy more of my overpriced bananas in return. Deal?")