Use an STM32 part equipped with DFSDM peripheral. Look at the ST website for those. DFSDM will accept the pulse density modulation and produce a usable result out of it. (STM32F412, STM32L476, etc..)
Otherwise you're left with an SPI and then processing it via software. Can be done on F103 I guess, was done before with MEMS microphones.
ST has a library for that, seen that somewhere, but I consider that to be stupid to do by SW.
If you have the AMC1203 I guess you do not need to cheap out, you probably will be able to buy a properly equipped MCU, like the STM32 with DFSDM. They are not as much more expensive compared to the F103 and also the F412 and L476 are both Cortex M4F, which will leave you a PLENTY of processing power for any DSP algorithms.