A recent conjecture of Di Francesco states that the number of domino tilings of a certain family of regions on the square lattice is given by a product formula reminiscent of the one giving the number of alternating sign matrices. These regions, denoted
T
n
\mathcal {T}_n
, are obtained by starting with a square of side-length
2
n
2n
, cutting it in two along a diagonal by a zigzag path with step length two, and gluing to one of the resulting regions half of an Aztec diamond of order
n
−
1
n-1
. Inspired by the regions
T
n
\mathcal {T}_n
, we construct a family
C
m
,
n
a
,
b
,
c
,
d
C_{m,n}^{a,b,c,d}
of cruciform regions generalizing the Aztec diamonds and we prove that their number of domino tilings is given by a simple product formula. Since (as it follows from our results) the number of domino tilings of
T
n
\mathcal {T}_n
is a divisor of the number of tilings of the cruciform region
C
2
n
−
1
,
2
n
−
1
n
−
1
,
n
,
n
,
n
−
2
C_{2n-1,2n-1}^{n-1,n,n,n-2}
, the special case of our formula corresponding to the latter can be viewed as partial progress towards proving Di Francesco’s conjecture.