When a $p$-dimensional parameter $\theta$ is defined through the moment condition $Em(X,\theta)=0$, a simple estimation procedure of $\theta$ was proposed in literature based on the unconditional expectation $Em(X,\theta)=EH(Z,\theta)$ for some function $H$, when X, a $k$-dimensional random vector, are contaminated with Laplace measurement error $U$, that is, only $Z=X+U$ can be observed. However, the estimation procedure was designed particularly for the cases where the components of the measurement error vector $U$ are independent. In this paper, we first introduce a general multivariate Laplace distribution, then extend the existing method to the general multivariate scenario. However, an example shows that the estimation procedure based on the unconditional expectation does not work in some cases. In this paper, we will propose an estimation procedure based on the condition expectation $E(m(X,\theta)|Z)$. Large sample properties of the proposed estimation procedure when $X$ is one-dimensional are discussed.