Sure sounds like a job for a DSP. Except DSP is a bit more difficult and expensive (though definitely better suited for that job...)
I guess it could be done with a microcontroller and some skill in analog electronics if you don't need to i.e. compress the data on the fly, but plain raw 8-16bit raw wav satisfies you. In this case I would take the following approach:
1) Microcontroller initializes the filesystem operation, opens a file for writing, initializes a fast ADC converter and sets the drive to DMA operation.
2) Whole operation goes outside the microcontroller: The drive uses DMA mode but doesn't access _memory_, just reads current/last ADC readouts as data and its address request are ditched. This removes all the harddrive operation overhead from microcontroller which may be used to synchronize the ADC, may work as a buffer/latch between it and the drive, wait for user input or such.
3) When done, microcontroller stops sending data and closes the file.
4) If you really want, use dedicated mp3 compressor chip and recompress the file from the drive while not recording.
Simply put reading analog input + processing it (compressing?) + controlling a filesystem + writing data to hard drive all at once is way too much for a microcontroller. If you "outsource" some of the work to external hardware it can work though...