Can every tromino (including those with gaps) tile the plane?

Each four-celled animal tiles the plane!

http://www.sciencedirect.com/science/article/pii/0097316585901050

The corresponding result for 1D three-celled "animals" holds as well. This was a recent problem in the German Math Olympiad (Problem 531046).

Together with Wolfgang's comment this solves the original problem, since if all coordinates of the three squares are different, then one starts by horizontal translation of the "animal" by all elements of $\mathbb Z$, and then uses the 1D result.