8~16 PDM digital microphone + what processing? = beamformer!
If you need very tight synchronisation/timekeeping, I think I'd go for an FPGA with microcontroller (either with e.g. ARM+FPGA on one chip, soft core, or external) and USB (or ethernet if you want)  

An medium size FPGA should be able to easily handle 16 channels in parallel, process to an 8 or 16-bit bus, buffer and send over to the micro using a parallel port (or maybe high speed SPI)
You can use a simple protocol between the FGPA and micro (if external), which sends e.g. 8 x 16-bit words at a time which were all captured simultaneously. You could add timestamps at the end of each chunk also.
Then send it all over to the PC via USB/Ethernet for processing.   

I'd look at what places like Digilent have to offer in the way of Dev boards, maybe a Spartan-6 board with an ARM Cortex-M3 or something.