The LEDs aren't directly hooked up to those signals. It's actually a little latch/address system with digital inputs, so you need push/pull signals. If you want to use diodes, you'll need to put a pull-down resistor (~1k to ground) on each line on the LED board side of the diode configuration, and then it's essential that the JAMMA switcher powers down the unused boards. Also, use type 1N4148 (or 1N914), not 1N4001.
However, the easiest way to do this is probably with some tri-state buffers. Handily, you've got exactly 8 signal wires (4 data, 3 address, and a latch signal), and buffers are readily available in an 8-wide configuration.
The best part number is probably a 74HC541. They're cheap (about 60 cents each), and you'll need 3 (one for each board). Just hook the 8 signal lines up to the inputs on each buffer, hook the outputs of the three buffers together and on to the LED board. Apply 5V/GND to the buffers and to the LED board (you can just re-use the 5V/GND that are part of that 12P harness).
You then need a way to signal the buffers. There are two enable lines, both active low, and they both need to be low for the buffer to be enabled. You only need one, so just tie one of them to ground. You then need a 5V logic signal for the other side. If your JAMMA switcher switches power you can just get a 74HC14, hook the 3 power lines up to three of the inputs (ground the other three) and use those as your enables for each buffer.
It may be a decent idea to put ~47ohm series resistors on the output of each buffer (before they all get tied together) to make sure things don't break if there's a short overlap where both are enabled.
Should be easily built on a piece of stripboard (aka veroboard) or perfboard.