Rather general versions of these theorems have been established for "sets of finite perimeter." You can find a recent paper on this subject here: http://www.math.northwestern.edu/~gqchen/10-Papers/ChenTorresZiemer.pdf

On the other hand, you say you don't want "heavy machinery," so it is hard to guess what you are looking for.

Rather general versions of these theorems have been established for "sets of finite perimeter." You can find a recent paper on this subject here:

Chen, Gui-Qiang; Ziemer, William P.; Torres, Monica, Gauss-Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws, Commun. Pure Appl. Math. 62, No. 2, 242-304 (2009). Available at arXiv:0709.3673 [math.AP]. Zbl 1158.35062.

On the other hand, you say you don't want "heavy machinery," so it is hard to guess what you are looking for.