I got a unit that shows some information on a 4*20 8bit HD44780 display. I'm not able to alter anything on the main MCU/LCD unit, but I'd like to use a arduino to sniff the data shown on the LCD and then send it to a PC over seriell.
you need to sample the lines clocked to your LCD, maybe its easier to use an 74HC165N - http://www.sparkfun.com/products/9519 - to capture the bits all at once iso doing it in code.
Because the 44780 is so slow, the MCU probably won't write to it at a rate that's too fast for the Arduino to read easily.
For a processor with a bus, I'd use a chip like an 8255 or 6522, because they can latch the data off the gadget's bus and signal you with an interrupt. You definitely need some sort of latch to catch the LCD data, and some sort of flip-flop to signal the Arduino that there's data to be read.
To be sure you get all the data, a small FIFO would be best. There used to be some relatively cheap 64-byte FIFOs available on the hobby market, but I haven't shopped for one in ages, so I don't know whether they're still easy to get. bgmicro.com is one of the few surplus dealers that carries much in the way of old 8080 and Z-80 family chips. I'd check there in addition to places like Mouser and Digi-key. Especially if they turn out to be expensive new.