Hi,
I would say analytic continuation is, in many cases, almost like black magic!
These problems are very, very hard. There is probably no sensible general answer. Not surprisingly, since solving the Riemann Hypothesis is trivially equivalent to the question of whether $1/\zeta$ has a continuation to $\{ \mathrm{Re}(z) > 1/2 \}$ or not!
Analytic continuation is very ill-posed: approximating a function $F$ by a sequence $F_n$ usually is of no use at all in determining the domain you can extend $F$ to. So you usually need exact formulae for your functions, expressed as infinite series, products, integrals or something else; numerical computations are almost certainly useless for these problems.
Often, existence of a continuation depends on cancellation in very complicated oscillating series or integrals, so you rarely have nice things like absolute convergence (or even conditional convergence!) on the true domain of the function. It can be very difficult to analyse the formulae you get.
Unfortunately you probably will just have to do lots of detailed algebraic calculations with the specific functions you are considering. Cauchy's theorem for functions expressed using contour integration is about the only general method I can think of, although it's not always possible.