Have you tried replacing the RAM. if the first part of the RAM it accesses is bad, it's not going to chime as it doesn't have enough memory to do the POST routine.

How much RAM do you have and how many modules? I would try that first. Also remove any other cards in it. Also run it without the G3 card.

My LC II did something like this, and I found out that the token-ring card in it was bad. I removed it, then found out the RAM module it had in it was bad. Replaced those and it works great now