So I thought, "Hey, why not take that idea and expand on it further?" In Chess, you don't have to be troubled with hunting down all of the other player's pieces because the only piece that really matters is the King. So what if we added a "king" piece to this rock-paper-scissors game? Perhaps the "king" can take any piece, but can also be taken by any other piece in the game. If you lose your "king", your game is over.

Adding a king piece would allow for some more strategy, faster games and less "wrap it up after you've already won" games, but it would still not give it a whole lot of extra depth, since the primary gameplay would still be the same. So lets add another element: taking multiple pieces at once.
(Additionally, in order to make sure that there is an even number for each of the rock, paper and scissor pieces, we're going to add a "defender" or "queen" piece in front of the "king", which acts the same way that the "king" does. It can take any piece, and be taken by any piece.)
Since the game will take place on a large checkers-style board, with more than enough empty tiles, we'll let the players take advantage of those empty spaces in their strategy. When your piece takes one of your opponent's pieces, your piece will take the other piece's spot. Now, if from that spot your piece can take another one of your opponent's pieces, then it can take that piece in the same turn. This can continue until your piece reaches a tile from which none of your opponent's pieces are vulnerable.

That means players can use sneaky combination attacks on their opponents. For example, lets suppose that you're the red player in the depiction above. Notice the paper where the green arrow starts. Using that piece of paper, you could take the neighboring rock. That would result in your opponent using his or her scissors to eliminate your paper, since it poses a threat to your opponent's "king". It would be a 1-for-1 deal, with neither player gaining any real advantage from it.
However, since your opponent has two rocks lined up, your piece of paper can take both the adjacent rock, and once it has taken that rock's spot, it can then take the next rock down the line as well. Plus, it is now next to your opponent's "king", which means that from three tiles away you can take your opponent's "king" and win the game. 3-for-0. A much better deal.
That new rule will add some more strategic depth to this game beyond the "my piece takes your piece, your piece takes my piece" mantra that it would otherwise consist of. It might be good to throw in a few additional rules, such as allowing any piece to move two tiles on its first move. Its just a matter of play-testing it out to see which way is more strategic and fun.
How about adding yet another piece: a wild "queen". The queen can take any piece if she attacks, but can also be taken by any piece if she is landed on. She takes on the quality of the tool used to destroy the first piece, so the queen can kill two papers or two scissors in a row but cannot kill a paper and a scissors in a row (alternately: she takes on the nature of the tool she destroys, so she could kill a paper, then kill a rock, then kill a scissors in a row).
Then the board could be 10x10, and you could have 20 pieces to a side, consisting evenly of six papers, six scissors, six rocks, a king, and a queen.