SPI transfer doesn't quite work on Netduino -> led driver
Referring to the datasheet, page 8:

VIH High-level input voltage is 0.7*Vdd minimum. Which is 3.5v! This is higher than the 3.3v signals you are sending it. You're working just beyond the operating spec of the part.

You can use a voltage converter to boost those 3.3v signals to 5v.

OR

Reduce the 5v supply voltage of the TLC5916 a little. You could easily do this by adding a diode to its VDD supply pin. This will drop a little voltage. Might just bring it into spec. Make sure that the diode's forward voltage drop is about 0.7v.