In this paper we study three new shifted sums of Apostol-Dedekind-Rademacher type. The first sums are written in terms of Jacobi modular forms, and the second sums in terms of cotangent and the third sums are expressed in terms of special values of the Barnes multiple zeta functions. These sums generalize the classical Dedekind-Rademacher sums. The main aim of this paper is to state and prove the Dedekind reciprocity laws satisfied by these new sums. As an application of our Dedekind reciprocity law we show how to derive all the well-known results on Dedekind reciprocity law studied by Hall-Wilson-Zagier, Beck-Berndt-Dieter, Katayama and Nagasaka-Ota-Sekine.