Some play by feel. Raymond Ceulemans has said that he won his first world 3-cushion championship without using any systems. of course he grew up playing straight rail and balkline so he pretty much knew every square millimeter of the table by heart. He later published a book on diamond systems.
As for the OP's question, if you want an accurate, coherent explanation of diamond systems, get Robert Byrne's books on pool and billiards. Then practice a lot.
Once again, Mr Jewett is being too modest.
Byrne's books are well worth having and contain a wealth of information,
but, who reads books anymore?
http://www.onthebreaknews.com/Jewett4.htm#Oct05
will take you to Bob Jewett's article about the most commonly
used diamond system for 3C. This will be more than enough to get you
started.
A quick Googling of "3 Cushion Billiards Diamond System" yeilds several
videos of real life.
My humble suggestion:
Start with what I call the "one pocket kick".
If you place the CB hung up in a foot pocket, shoot at aprox the second
diamond on the opposite long rail, you should scratch in the other foot
pocket.
This is the basic corner-to-corner kick. Once mastered, it is a good basic
foundation to learn all the possible paths.
A few have pointed out that this is theory - and the real world will be more
complicated. True enough, but that is the case with any system.
It will take time and lots of practice to develop consistent skill -
IMHO - well worth it.
Dale