How to Minimally Communicate with an ADS131E04 ADC via SPI
Are you sending an additional transfer to allow the part to send data back? The SPI protocol does not allow for the slave device to send back data without the master sending the necessary clocks. To send these clocks, you need to send another transfer. The process is like this:


The master selects a slave (~CS goes low).
The master sends a request for data to the slave (MOSI->DIN, SCLK clocks).
When the transfer completes, the slave decodes the request and puts the data in its transmit buffer.
The master sends a "dummy" byte of data to the slave  (MOSI->DIN, SCLK clocks).
During that "dummy" transfer, the slave returns the requested data to the master (DOUT->MISO).
The master removes the data from its receive buffer and releases the slave (~CS goes high).


I suspect that you are missing that fourth operation.