Have you tried to determine the states of A & B that will produce Y? After that, it's simple to work in the C input. It's helpful if you label the gates, too (U1, U2, etc.).
Start from the list with all possible combinations of A, B and C. Then work your way from the left to the right of the circuit. For given A and B find the output of the first gate. Then use it along with B to find the output of the second gate and so on.
Basically what has been said previous. Just make and use a truth table. If you need visual assistance draw the circuit 8 times, and use a red or green magic marker to record the output of each gate to fill the truth table.