Tile away!

First off, since the 3x1 tiles only fit in one direction, the two 1x10 rows can be taken independently, and the problem boils down to figuring out how many ways a 10x1 grid can be tiled and then squaring it.

Let $N_n$ be the number of ways to tile an nx1 grid.

For a 10x1 grid, the far left tile can either be a 1x1 or a 3x1, and the rest can be filled in either $N_7$ or $N_9$ ways. In other words:

$N_{10} = N_9 + N_7$

To generalize this, for $n > 2$ ,

$N_n = N_{n-1} + N_{n-3}$

And clearly,

$N_0 = N_1 = N_2 = 1$

Solving,

$N_{10} = 28$

So, the total number of ways to tile a 2x10 grid is:

$N_{total} = N_{10}^2 = \boxed{784}$