As Jason said already, there are many
examples of Moishezon manifolds which
are rationally connected. Indeed, any
manifold bimeromorphic to a rational
connected manifold is again rationally
connected, and there are many non-Kahler
Moishezon manifolds bimeromorphic to
$CP^n$ (or any other Fano manifold,
I suppose).

The better question would be whether all
rational connected manifolds are Moishezon.
The answer is, suprisingly, positive,
if you define rational connectedness
one way, and negative for another definition.

Recall that a rational curve is called
**ample** if its normal bundle
is $\bigoplus O(i_k)$ with all $i_k$ positive.

The first definition is

**Defintion:**
A compact complex manifold $M$ is called **
rationally connected** if there is an ample
curve with compact deformation space (Barlet
or Douady, does not matter).

In this case, any rationally connected manifold
is Moishezon, as shown by Campana

F. Campana, *Reduction algebrique d'un morphisme
faiblement Kahlerien propre,* Math. Ann. 256 (1980), 157--189.

Another definition is the same, but without the compactness
assumption. In this case you have manifolds which
are rationally connected, but not Moishezon, for example,
the twistor space of K3 surface or a torus. Curiously,
there is a rational curve passing through every
$n$ points of such a manifold; this is explained,
for example, in my paper
"Rational curves and special metrics on twistor spaces".

Personally, I think the first definition makes more sense.