Generalizing the algorithm for domino tiling?

Question

In this earlier question the OP asked the following problem:

Given a rectangular grid where some squares are empty and some are fille

Answer 1

1x3 tiles are hard by reduction from cubic planar monotone One-in-three 3SAT. We have to build some "circuitry" to encode the formula.

"Gates":

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X********Y

Forces exactly one of X and Y to be covered externally. Used to link a variable and its negation.

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Y***
   *
   *
  ooo  ****
  * *  *  *
  * *  *  *
  X ****  Z

Forces none or all of X and Y and Z to be covered externally. Used to copy X or destroy three copies of the same thing. Wires can be shaped more or less arbitrarily using length-3 L pieces.

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*******************
*        *        *
*        *        *
X        Y        Z

Forces exactly one of X and Y and Z to be covered externally. One for each clause.