The center of mass of each rod is right in the middle of the rod. One is at (L/2, 0), the other at (0, L/2). So, for the purpose of finding the center of mass of the entire structure, you can treat the mass of each rod as being right at those points. Where's the center of mass of two equal masses?
The way that I approached it was picking the point (L,0) and calculating the moments about this point using the centres of mass of the two separate rods (M*0.5*L+M*L) this has to be equal to the total mass acting at the centre of mass x coordinate (2.0*M*x). x is found to be 0.75*L and we have our answer.