For a cyclic group G acting on a smooth variety X with only one character occurring in the G-equivariant decomposition of the normal bundle of the fixed point locus, we study the derived categories of the orbifold {[}X/G] and the blow-up resolution (Y) over bar -> X/G. Some results generalize known facts about X = A(n) with diagonal G-action, while other results are new also in this basic case. In particular, if the codimension of the fixed point locus equals vertical bar G vertical bar, we study the induced tensor products under the equivalence D-b((Y) over bar) congruent to D-b({[}X/G]) and give a `flop-flop = twist' type formula. We also introduce candidates for general constructions of categorical crepant resolutions inside the derived category of a given geometric resolution of singularities and test these candidates on cyclic quotient singularities.