That's a tough question. Every civilization has passed on it's legacy before it succombed to it's problems. I mean... the very first 'civilization' (if you can even call it that) of humans which consisted only of small tribes building fires at night certainly contributed something important-- the use of fire.

Next came the use of stones...

Then metals...

The ancient Greeks and Romans gave us the scientific method, basic military philosophy, and many other things that have already been mentioned.

The ancient Greeks had flamethrowers and a substance similar to napalm. They even had a mechanical device which could measure distances. They (or the Romans, I think) later used it to help in marking their roads accurately.

The middle ages, while being dark, gave us water wheels and I think even windmills (but I'm not positive about that).

It's hard to say.

As for the greatest civilization? Well I vote for the one. That includes everyone, not just the West.

Modern civilization has given us a comfortable middle class, the beginings of space travel, the first real understanding of our world, and world-wide communication. It's brought many dangers, but it also offers much hope. This may be the first civilization to full secure humanity's future by colonizing space. (and thus, giving birth in effect to a whole new one)