Not all cyclic groups are created equal; some groups are used (currently) for cryptographic applications, while others are not.
"Groups of cryptographic interest" refers to groups that are normally used in actual applications of cryptography that relate to the discrete log problem; these are Diffie-Hellman key exchange, ElGamal, and the like.
Usually, the groups that are used for these problems are the multiplicative group of integers modulo a very large prime $p$; the multiplicative group of a finite field (particularly of finite fields of characteristic $2$, because they tend to be easy to implement); or points of curves of elliptic curves (over finite or global fields).
Some groups have very easy discrete logarithm problem (the additive group of integers modulo $n$, for example) so they are not used in actual applications of cryptography; other groups are too hard to implement, so they are not used either. The discrete logarithm problem for these groups is irrelevant for cryptography, since they are not used for cryptography. So these groups are not of "cryptographic interest".
Note that being of "cryptographic interest" is both time-dependent (it depends on what is being used now), and more importantly as noted by Qioachu, it is not invariant under isomorphism. The multiplicative group of a finite field of order $p^k$ is (abstractly) isomorphic to the additive group of integers modulo $n=p^k-1$; but while the discrete logarithm problem for the former is considered "hard", the discrete logarithm problem for the latter is "easy". The problem here is that finding an isomorphism is essentially equivalent to constructing a full logarithm table for the multiplicative group of the finite field, so having an isomorphism is pretty much the same as solving the discrete logarithm problem.