if you are looking for a development board to start the project Microchip's microstick could be useful
**broken link removed** YouTube - Microchip Microstick Overview
it has an onboard programmer and header to connect to a breadboard for your own circuits
I used PIC18F4620 earlier to develop a product which is used to read data from SD/MMC and displaying on GLCD and also
stores a txt file which we typed on GLCD ....
I was strucked interfacing with SD/MMC with PIC18F4620...
I use 18F4620 for interfacing with an microSD card, and I don't have any problems with it.
The problem is not with the mcu but rather with the hardware or software you are using. Most problems with sd cards usually occur due to improper signal level conversion/handling.