even a ddr 3 board that would support your e6400 wouldnt support i3.
also, motherboards supporting core2 and ddr3 are probably considerably overpriced at this point. the hardware is obsolete.
your best bet is to get a new cpu, motherboard and ram all in one go.
if you are having problems finding something in your price range you might consider looking on ebay for something from the sandybridge generation.