Short Answer: There is no Linux driver for this device yet.
The device you're actually referring to (the 7-in-1 memory card reader), is this device:
Mass storage controller: Texas Instruments PCIxx21 Integrated FlashMedia Controller
The device you've referred to is the PCMCIA/ CardBus controller chip, which is supported.
With regards to the 7-in-1 FlashMedia controller - it uses a proprietary Texas Instruments interface, so there is currently no available driver, and no easy way for anyone to write one.
There are a few projects around trying to reverse engineer the Windows driver and produce a Linux one, though only one is currently active: http://tifmxx.berlios.de
(note that the project author is only working towards using SD/MMC cards under Linux at the moment).