Parallel hash construction
Dmitry said that a collision in the xor can be found with a generalized birthday attack.

Actually, it's a lot worse than that; second preimages of sufficiently long messages can be found in polynomial time; hence an enormous block size (which might help against the birthday attack) doesn't help.

Let us call the size of the output of the ideal block cipher $n$.  Then, let us consider a message that consists of (at least) $n$ blocks; and for each block, we consider two distinct alternatives, at position $i$, we would have arbitrary selected blocks $M_i$ and $M'_i$.

We would then compute $H_i = E_i( M_i ) \oplus M_i$ and $H'_i = E_i( M'_i ) \oplus M'_i$.

If we have a target value $Z$ for the exclusive or's (which we would have for a second preimage problem), we would then consider what sum of $H_i$ and $H'_i$ would exclusive-or to the target values; this can be viewed as a linear algebra problem, and can be solved with, say, Gaussian elimination in time $O(n^3)$