There is the cheapest way, and then there is the quickest way. Cheapest way would be to pick a flavor of microcontroller ATMEL/PIC/etc, build a programmer, download the programming software, go through the steep learning curve of learning the ins and outs of getting it running.
If this is just a one-off or a prototype and you just want to get it done without turning it into a full-time job, I'd suggest just buying a Basic Stamp and going from there. I just took a quick look at the ISD1700 datasheet, and I wouldn't even bother trying to interface via SPI if it wasn't necessary. Since you are only playing four messages, but can only play stored messaged sequentially (like many of these ISD chips, this one was meant to be used in answering machines), I would think you could get away with just pulsing the PLAY and FWD lines.
I could see it going something like this with a Basic Stamp (or any other microcontroller).
1) Monitor your four inputs.
2) Input three is detected, so message three needs to be played.
3) Microcontroller pulses the PLAY line to start playback of the first recording, then quickly pulses the FWD line twice to play the third message. This all happens within a millisecond or two, so it should be seamless.
4) Lather. Rinse. Repeat.
A Basic Stamp makes something like this so trivial. You could breadboard and program something like this rather quickly. Your code would look something like this (been a while since I used the BS, but it's in the ballpark):
if input3=0 then gosub playmessage3 ' if input three is low, go to routine to play third message
playmessage3:
low PLAY ' low pulse on the PLAY line
high PLAY
low FWD ' two negative pulses on the FWD line
high FWD
low FWD
high FWD
return ' go back to checking your inputs
**broken link removed**
**broken link removed**