We study the arithmetically Cohen-Macaulay (ACM) property for finite sets of points in multiprojective spaces, especially
(
P
1
)
n
(\mathbb P^1)^n
. A combinatorial characterization, the
(
⋆
)
(\star )
-property, is known in
P
1
×
P
1
\mathbb P^1 \times \mathbb P^1
. We propose a combinatorial property,
(
⋆
s
)
(\star _s)
with
2
≤
s
≤
n
2\leq s\leq n
, that directly generalizes the
(
⋆
)
(\star )
-property to
(
P
1
)
n
(\mathbb P^1)^n
for larger
n
n
. We show that
X
X
is ACM if and only if it satisfies the
(
⋆
n
)
(\star _n)
-property. The main tool for several of our results is an extension to the multiprojective setting of certain liaison methods in projective space.