Color this grid in the following way. Then it it is easy to see that each $3\times 1$ piece will either cover $3$ grids of the same color (call such piece as a same color piece) or $3$ different colored grid (call such piece a three-color piece). The L-shape piece will always cover two grids of one color and one grid of another color. Let the number of three-color pieces be $m$. Also assume that there are $n_1$, $n_2$ and $n_2$ same color pieces to cover the three different colored grids. Without loss of generality, let's assume the L-shaped piece covers two grids of color 1, and one grid of color 2. Then, the number of grids of each color covered are: \begin{align} \text{Color 1:}\qquad&m+3n_1 + 2\\ \text{Color 2:}\qquad&m+3n_2 + 1\\ \text{Color 3:}\qquad&m+3n_3 \end{align} By the construction, the number of different colored grids are the same. Therefore, $$m+3n_1+2=m+3n_3\implies 3n_1+2=3n_3$$ This relation cannot hold because one is a multiple of $3$ and the other is not.