Bayesian estimation with other loss functions than the standard hit-or-miss loss or the quadratic loss often yields optimal Bayesian estimators (OBEs) that can only be formulated as optimization problems and which have to be solved for each new observation. The contribution of this paper is to introduce a new parametric family of estimators to circumvent this problem. By restricting the estimator to lie in this family, we split the estimation problem into two parts: In a first step, we have to find the best estimator with respect to the Bayes risk for a given nonstandard loss function, which has to be done only once. The second step then calculates the estimate for an observation using importance sampling. The computational complexity of this second step is therefore comparable to that of an MMSE estimator if the MMSE estimator also uses Monte Carlo integration. We study the proposed parametric family using two examples and show that the estimator family gives for both a good approximation of the OBE.