  In this paper, we introduce the notion of completely non-trivial module of a
Lie conformal algebra. By this notion, we classify all finite irreducible
modules of a class of $\mathbb{Z}^+$-graded Lie conformal algebras
$\mathcal{L}=\bigoplus_{i=0}^{\infty} \mathbb{C}[\partial]L_i$ satisfying $
[{L_0}_\lambda L_0]=(\partial+2\lambda)L_0,$ and $[{L_1}_\lambda L_i]\neq 0$
for any $i\in \mathbb{Z}^+$. These Lie conformal algebras include Block type
Lie conformal algebra $\mathcal{B}(p)$ and map Virasoro Lie conformal algebra
$\mathcal{V}(\mathbb{C}[T])=Vir\otimes \mathbb{C}[T]$. As a result, we show
that all non-trivial finite irreducible modules of these algebras are free of
rank one as a $\mathbb{C}[\partial]$-module.
