In this article, we study the problem of modifying smooth, algebraic stacks with finite, diagonalisable stabilisers such that their coarse spaces become smooth. The only modifications used are root stacks and blow-ups in smooth centres. If the generic stabiliser of the original stack is trivial, the canonical map from the resulting stack to its coarse space is also a root stack. Hence we can think of the process as removing stackiness from, or destackifying, a smooth stack with help of stacky blow-ups. The construction work over a general base and are functorial in the sense that they respect base change andcompositions with gerbes and smooth, stabiliser preserving maps. As applications, we indicate how this can be used for destackifying general Deligne-Mumford stacks with finite inertia in characteristic zero, and to obtain a weak factorisation theorem for such stacks. Over any field, the method can be used for desingularising locally simplicial toric varieties, without assuming the presence of toroidal structures.