If the group has a small number of elements you can check that no element
has order equal to the order of the group. If this is impractical, you can
show that the group has some property that cyclic groups do not have.
Or, you can show that cyclic groups have some property that the group does 
not. 
For example, a finite cyclic group can have at most one element of order
2. So, if you find two elements of order 2 the group is not cyclic.
Similarly, a finite cyclic group has at most one subgroup of any
particular order.  So, if you can find two subgroups of the same order the
group is not cyclic.