(1) A commutative Noetherian ring is reduced if and only if it is generically reduced
(i.e. $R_0$, i.e. regular after localization at all height zero primes) and $S_1$
(i.e. every prime of height at least one has depth at least one).

Since a ring is Cohen--Macaulay iff it is $S_i$ for all $i$, to construct a non-CM ring,
it suffices to construct a non-$S_1$-ring, and by the above, for this it suffices to
construct a ring which is generically reduced but not reduced (or more geometrically speaking,
has embedded components).

E.g. $k[x_1,\ldots,x_n,y]/(x_1 y, \ldots , x_n y, y^2)$ is such a ring, and has dimension $n$.
(To get a local example, localize at $(x_1,\ldots,x_n,y).$)

[Added: misread question; these are non-CM rings. For CM, but non-Gorenstein rings,
see Hailong Dao's answer.]

(2) Typically the easiest way to recognize that a ring is Cohen--Macaulay is to use the following facts: any regular local ring is CM; if $A$ is CM and $f$ is a regular element
of $A$ (i.e. a non-unit and non-zero divisor) then $A/fA$ is CM.

Arguing inductively, we find that if $f_1,\ldots,f_n$ is a regular sequence in a regular
local ring $A,$ then $A/(f_1,\ldots,f_n)$ is CM. In particular, complete intersections
in an affine space are CM (and thus so are any of their localizations).

(In fact, these rings will be Gorenstein, not just CM.)

In small dimensions, one can also use the fact stated in part (1) to conclude that a one-dimensional reduced Noetherian ring is CM, and one also has Serre's criterion $R_1 + S_2$
for normality, which shows that a normal ring of dimension two is CM.