1. e4 e5 2. Nf3 Nc6 3. c3 - don't see why no one ever plays the Ponziani!

I can't wait til Ponz posts on this one.

LOL

Seems like a good opening to me.

3...d5! is what's wrong. :P

Chessexplained did a video on the Ponziani for Black a couple of years ago:

https://www.youtube.com/watch?v=P0kXktxZyHs

This is hardly a comprehensive survey of the Ponziani, but it will give you an idea of what to avoid with White. Likely White has a path to equality somewhere, but Black can play for an advantage in most lines and White is just losing in the really sharp lines. It is a tricky opening, but I wouldn't make it my main choice against 1.e4 e5. However, one could argue that it hardly matters below 2000 since opening preparation is not so well developed:

A great opening for beating sub-2000 players

https://www.youtube.com/watch?v=JOZLHeKWhTQ

But in my opinion, the Ponziani really stinks for White! I'm really quite happy to face it with Black (but I'm not sub-2000 either).

It's premature without a preliminary 3.Bc4.  Like MetalRatel said 3...d5! is very strong.  Probably even known since the 17th century since we don't see Greco or even NN play the Ponziani, it's usually almost always 3.Bc4, presumably because 3...d5! is strong and 3.Bc4 develops and prepares castling anyway.