As poncho notes in the comments above, viewing the input block as a 128-element vector over ${\rm GF}(2)$, the AES ShiftRows and MixColumns operations are both linear transformations, and AddRoundKey is just vector addition.
Both linear transformations and addition of a constant are kinds of affine transformations, and since the composition of any two affine transformations is itself an affine transformation, it follows that, if the non-affine SubBytes step is removed from AES, the whole cipher becomes affine.
In particular, every affine transformation can be represented in the form $c = Ap + k$, where $p$ is the plaintext input (a vector of 128 bits), $c$ is the corresponding ciphertext output, $k$ is a constant vector and $A$ is a 128 × 128 bit matrix. Furthermore, the inverse of this transformation is then simply $p = A^{-1}(c - k)$, where $A^{-1}$ is the matrix inverse of $A$, and (for vectors over ${\rm GF}(2)$) both vector addition "$+$" and subtraction "$-$" simply mean bitwise XOR.
The constant $k$ obviously depends on the key, but the matrix $A$ actually does not — it is fully determined by the ShiftRows and MixColumns steps, neither of which are key-dependent. Thus, you can precalculate it e.g. by implementing this affine AES variant yourself, leaving out the AddRoundKey step as well to make it linear, and using this to encrypt (or decrypt) all the 128 blocks with a single bit set to one and all other bits zero, which will directly give you the columns of $A$ (or of its inverse $A^{-1}$).
Once you've precalculated $A$, even just a single known plaintext/ciphertext block pair $(p,c)$ encrypted using the actual affine cipher will let you determine the key-dependent the additive constant $k = c - Ap$. Knowing $A$ and $k$ will then let you encrypt (or decrypt, using $A^{-1}$) any arbitrary plaintext (or ciphertext) block.
c
sitting in the relationship); this is actually pretty easy to workaround; one way is adding an extra implied bit to each plaintext/ciphertext, which is set on every plaintext/ciphertext you have, and solve it as a set of GF(2) linear equations over 129 variables. $\endgroup$