Why is multiplication uncommon in cryptographic primitives?
There's at least 5 reasons why multiply is not more often used in symmetric ciphers and hashes:


For use as mixer, multiplication requires more hardware/energy/time than other hardware constructs of comparable cryptographic interest. This is an argument mostly for hardware implementations, but many ciphers (and a few hashes) are designed so as to be fast in hardware (that was the explicit implementation target for DES, and an explicit target for AES, which often is in hardware nowadays).
It is far from universal (and used to be uncommon) that wide multiplication is single-cycle, and the number of cycle(s) listed might not be the end of the story; often there is latency, and that cost extra cycles if the result is used for the next few instructions (for x64 otus pointed this source, the same applies to some ARM).

Worst there is often timing dependency on one operand (opening an avenue for timing attacks), and no way to control which operand from the comfort of a high-level language (other than by forcing some high-order bits in both operands). For example, the common 32-bit ARM Cortex M3 has UMULL (32x32->64-bit result) documented as requiring 3 to 5 cycles, with


  early termination depending on the size of the source values.

Multiplication only does bit mixing to the left: if $C=A\;B$ then bit $C_j$ is independent of bits $A_i$ and $B_i$ for any $i&gt;j$. Fixing this requires computing the product with more precision than its operand (and some post-processing), which might not be fast/easy in high-level language.
Multiplication by a constant is linear, and ciphers require non-linearity. While some derivatives of multiplication (like squaring) are not linear, multiplication as the sole source of non-linearity could open to algebraic attacks.
As pointed in  comment, when truncating the result of a multiplication (effectively working modulo some power of two), $F_A: B\mapsto F_A(B)=A\;B\bmod 2^k$ is a bijection only for odd $A$, and can loose a lot of entropy for other $A$, which is undesirable in a normal mixer. 


The only widely used cipher I can name that heavily uses multiplication other than by constant is IDEA, where it is followed by modular reduction modulo the prime $2^{16}+1$, in order to fix issues 3 and 5.

Multiplication is in wide use in asymmetric cryptography: RSA, DSA, Elliptic Curve cryptography over a prime field..