There are two naive approaches to Feynman integrals -- finite-dimensional approximations and analytic continuation. Both are discussed in S. Albeverio, R. Hoegh-Krohn, S. Mazzucchi, Mathematical theory of Feynman path integrals, Springer LNM 523, 2 ed. As far as I understand, both work fine as long as $L$ is a non-degenerate quadratic function plus a bounded one. Moreover, in the 1st edition of the above-mentioned book Albeverio and Hoegh-Krohn give yet another definition, which works under the same hypotheses.
Finally, let me mention yet another approach to Feynman integrals: the "white noise analysis" (see e.g. Lectures on White noise functionals by T. Hida and S. Si). It was used by S. Albeverio and A. Sengupta (Comm Math Phys, 186, 1997) to define rigorously the Chern-Simons integrals when the ambient manifold is $\mathbf{R}^3$ (they have to use gauge-fixing though). Since I don't know much about this I'd like to ask a couple of naive questions: Is there a way this could help one handle the cases other approaches can't? Can this help eliminate the necessity to start with a non-degenerate quadratic function?