You can connect an SD card to some ESP8266 with the SPI GPIO broken out. For example: ESP8266 201 or 12E
Then connect your sd card (FAT32) like this to the ESP:
Take these instructions for help: http://www.esp8266.com/viewtopic.php?f=32&t=3558#p20419
Be sure to change CS to a free GPIO. Normally, this would be IO15 which has to be pulled to ground. Otherwise the ESP won't work. In this case I take IO2.
Now you can run the example “CardInfo” sketch included by the Arduino IDE. (BE sure to change the chip select pin!)
Do you have any questions? Feel free to ask here: http://www.esp8266.com/viewtopic.php?f=32&t=3558