I rediscovered my book from Simon Singh ´The Simpsons and Their Mathematical Secrets´ and finished it.
I really liked the chapter which included the nontransitive dice and I wanted to have some of these. So because there where non to buy or to find that I liked, I just made some as they are described in the book.
These were my favorites because of the triple system as you have it with RPS and also the chances are the same with each combination. So not the same chance like 50:50 but independent from choices there is no best dice.
Chances are the following:

A vs. B - A wins 20 of 36 possible combinations
B vs. C - B wins 20 of 36 possible combinations
C vs. A - C wins 20 of 36 possible combinations

So everytime you choose second you may have a chance of 56% winning.
Just if you wonder about the numbers, they are like this:

Dice A: 3, 3, 5, 5, 7, 7
Dice B: 2, 2, 4, 4, 9, 9
Dice C: 1, 1, 6, 6, 8, 8

Hope you like it, so feel free to try your luck and roll the dice. :-)