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We study the phase diagram of
Q-state Potts models, for
Q
=
4
cos
2
(
π
/
p
)
a Beraha number (
p
>
2
integer), in the complex-temperature plane. The models are defined on
L
×
N
strips of the square or triangular lattice, with boundary conditions on the Potts spins that are periodic in the longitudinal (
N) direction and free or fixed in the transverse (
L) direction. The relevant partition functions can then be computed as sums over partition functions of an
A
p
−
1
type RSOS model, thus making contact with the theory of quantum groups. We compute the accumulation sets, as
N
→
∞
, of partition function zeros for
p
=
4
,
5
,
6
,
∞
and
L
=
2
,
3
,
4
and study selected features for
p
>
6
and/or
L
>
4
. This information enables us to formulate several conjectures about the thermodynamic limit,
L
→
∞
, of these accumulation sets. The resulting phase diagrams are quite different from those of the generic case (irrational
p). For free transverse boundary conditions, the partition function zeros are found to be dense in large parts of the complex plane, even for the Ising model (
p
=
4
). We show how this feature is modified by taking fixed transverse boundary conditions.