CodenameHaswell :
Hence why I said "if the motherboard supports it".
Indeed. That statement implies that an i3 or better can be inserted into the current motherboard (the OP did state "without changing motherboard.") In addition your phrase "new cpu" is ambiguous, an individual may believe it to be a brand new Core 2 Duo/Quad, or other processor.
I'm not "hating" on you, facts need to be made clear in my opinion.
In response to the original poster, no you cannot upgrade to an i3 or above on your current motherboard if you currently have a Core 2 Duo. The Core 2 Duo and Quad processors are on Intel's socket 775 for desktops. The more modern Intel CPU's such as the i3 are on a different socket, the most recent being 1150 for general consumers.
With that said, if your RAM is currently DDR2 (most likely if you're on a Core 2 Duo) then you'll need a new motherboard and RAM (you'll need DDR3 RAM.)
All the best.