How to read data using Arduino SPI

You call SPI.transfer() four times, each time saving the returned value to a variable after proper bit shifting.

uint32_t val;
val = SPI.transfer(0xff);  //0xff is a dummy
val |= (uint32_t)SPI.transfer(0xff) << 8;
val |= (uint32_t)SPI.transfer(0xff) << 16;
val |= (uint32_t)SPI.transfer(0xff) << 24;

I assume the least significant byte is received first. Make sure the SPI mode is the right one, as indicated in your datasheet.

If the slave can't handle being deasserted between bytes (which is what SPI.transfer() will do at the end of the transaction), then you can either try st2000's hardware SPI approach or use bitbanged SPI with shiftIn().

Sparkfun has a good explaination of what SPI is:
https://learn.sparkfun.com/tutorials/serial-peripheral-interface-spi

The Arduino IDE has a SPI library that comes with the IDE.
http://www.arduino.cc/en/Reference/SPI

The library comes with two examples:
Using SPI to read a Barometric Pressure Sensor
Controlling a Digital Potentiometer Using SPI

Objective is to read 32 bits using the (unknown) device's SPI port.

If the device will tolerate the SPI Chip Select line activity (going from inactive to active to inactive for each 8 bit byte read) you should be able to get the desired 32 bits of data by performing 4 consecutive 8 bit SPI reads.

However, if the device will not tolerate the above SPI Chip Select line activity (that is, if the device requires the SPI Chip Select to be active for the entire 32 bit SPI transaction), you can not separate the 32 bit transaction into 4 individual 8 bit SPI transactions with out controlling the SPI Chip Select line. The following code example illustrates how this is done:

void loop(){
//transfer 0x0F to the device on pin 10, keep the chip selected
SPI.transfer(10, 0xF0, SPI_CONTINUE);
//transfer 0x00 to the device on pin 10, keep the chip selected
SPI.transfer(10, 0×00, SPI_CONTINUE);
//transfer 0x00 to the device on pin 10, store byte received in response1, keep the chip selected
byte response1 = SPI.transfer(10, 0×00, SPI_CONTINUE);
//transfer 0x00 to the device on pin 10, store byte received in response2, deselect the chip
byte response2 = SPI.transfer(10, 0×00);
}

The above example comes from the Arduino SPI library web page and is a 16 bit (not 32 bit) transfer. To code up a 32 bit transfer, continue to call the SPI.transfer method using the SPI_CONTINUE parameter.

added later...

It appears people are discouraging use of the DueExtendedSPI methods. And SPISettings and SPI.beginTransaction() are to be used? If so this page show example code where the code is explicitly controlling the SPI Chip Select line (look for slaveAPin & slaveBPin). Note how the code is reading 24 bits from slave A and writing 8 bits to slave B. To read 32 instead of 24 bits, we need to alter a segment of the code (along with a few other supporting changes) in this fashion:

...
  SPI.beginTransaction(settingsA);
  digitalWrite (slaveAPin, LOW);
  // reading only, so data sent does not matter
  val0 = SPI.transfer(0);
  val1 = SPI.transfer(0);
  val2 = SPI.transfer(0);
  val3 = SPI.transfer(0);
  digitalWrite (slaveAPin, HIGH);
  SPI.endTransaction();
...