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The subject of this paper is the index theorem on odd-dimensional manifolds with boundary. Such a theorem has been formulated and proved by D. Freed and his proof is based on analysis by Calderon and Seeley. In this paper we prove this theorem using the heat kernel methods for boundary conditions of Dirichlet and Neumann type. Moreover, we also consider the Atiyah-Patodi-Singer spectral boundary condition which is not studied in Freed's paper. As a direct consequence of the method, we obtain some information about isospectral invariants of the boundary conditions. This proof does not use the cobordism invariance of the index and is generalized easily to the family case. [ABSTRACT FROM AUTHOR]