I'm not sure if the wiring colours are the same on the MX-3 as the 323F BA, but you need to find IG1 and START. The relay needs to bridge those together.
Sounds like you've wired the relay wrong!
[img]http://www.steelnthings.com/pins1.gif[/img]
You need to go from a 12v feed (i'd run one directly from the battery, fused), into the start button switch, then into pin 86 on the relay. 85 then goes to ground.
You then need to connect pin 87 to either IG1 or START, then pin 30 to the other one of those two.
EDIT:
According to
http://www.modifiedlife.com/1992-mazda- ... schematic/
You need the black/blue wire, and the blue wire, the same as the 323F BA I think
