A fundamental problem in wireless networks is the \emph{minimum spanning tree} (MST) problem: given a set $V$ of wireless nodes, compute a spanning tree $T$, so that the total cost of $T$ is minimized. In recent years, there has been a lot of interest in the physical interference model based on SINR constraints. Distributed algorithms are especially challenging in the SINR model, because of the non-locality of the model. In this paper, we develop a fast distributed approximation algorithm for MST construction in an SINR based distributed computing model. For an $n$-node network, our algorithm's running time is $O(D\log{n}+μ\log{n})$ and produces a spanning tree whose cost is within $O(\log n)$ times the optimal (MST cost), where $D$ denotes the diameter of the disk graph obtained by using the maximum possible transmission range, and $μ=\log{\frac{d_{max}}{d_{min}}}$ denotes the "distance diversity" w.r.t. the largest and smallest distances between two nodes. (When $\frac{d_{max}}{d_{min}}$ is $n$-polynomial, $μ= O(\log n)$.) Our algorithm's running time is essentially optimal (upto a logarithmic factor), since computing {\em any} spanning tree takes $Ω(D)$ time; thus our algorithm produces a low cost spanning tree in time only a logarithmic factor more than the time to compute a spanning tree. The distributed scheduling complexity of the spanning tree resulted from our algorithm is $O(μ\log n)$. Our algorithmic design techniques can be useful in designing efficient distributed algorithms for related "global" problems in wireless networks in the SINR model.