In this article we show that random walks in random environment on $\mathbb{Z}^d$, $d \ge3$, with transition probabilities which are
$\varepsilon$-perturbations of the simple random walk and such that the
expectation of the local drift has size bigger than $\varepsilon^\rho
$, with $\rho< \frac{5}{2}$, when $d=3$, $\rho< 3$, when $d \ge4$, fulfill
the condition (T$^\prime$) introduced by Sznitman [Prob. Theory Related
Fields (2002) 122 509-544], when $\varepsilon$ is
small. As a result these walks satisfy a law of large numbers with
nondegenerate limiting velocity, a central limit theorem and several
large deviation controls. In particular, this provides examples of
ballistic random walks in random environment which do not satisfy
Kalikow's condition in the terminology of Sznitman and Zerner [Ann.
Probab. (1999) 27 1851-1869]. An
important tool in this work is the effective criterion of Sznitman.