USB / Serial converter without FTDI Chip
Yes you can make a native USB device without the FTDI IC.

Some ICs from Atmel (ATtiny2313) an Microchip (PIC18F2550) has the USB ports native. So its just plug and play.

This site has some good examples to use ATtiny, and this one with PICs. 

All of them use the V USB to comunicate with computer, emulating a Serial port on it.