The orbital period certainly depends on the mass of the black hole - this is present in the formula above in a disguised form, as the Schwarzschild radius r_s of the black hole.
r, the Schwarzschild radius of the orbit, can be losely thought of as "distance from the black hole", but it should not be confused with the distance that one would measure with a ruler. The defintion of r is that the circumference of a circle drawn around the black hole at any Schwarzschild coordinate r is 2 \pi r
The formula doesn't depend on the density or volume of the black hole, which is not defined. The formula also does not depend on the mass of the object that is orbiting as long as the object orbiting is small. Similarly the density of the orbiting object doesn't matter, either.
The formulas needed can be found in various textbooks, or online at
http://www.fourmilab.ch/gravitation/orbits/
Actually the source above has worked a fair bit of the problem, something
I didn't realize when I posted the question. Anyway
The differential equations that describe the orbit in Schwarzschild coordinates r and \phi, both functions of proper time \tau, are given by:
<br />
(\frac{dr}{d\tau})^2 + V^2(r) = E<br />
<br />
V^2(r) = (1-\frac{2M}{r})(1 + \frac{L^2}{r^2})<br />
<br />
\frac{d\phi}{d\tau} = \frac{L}{r^2}<br />and another equation for coordinate time which we won't need. In the above expression, E and L are conserved quantites, corresponding to energy and momentum (actually energy / unit rest mass and momentum per unit rest mass). M is the mass of the black hole. Geometric units are being used at this point, i.e. G=c=1.
These equations give us enough information to solve for r(\tau) and \phi(\tau).
However, its not immediatly obvious how to solve the equations for the conditions that an orbit be circular. The tricky bit is realizing that circular orbits occur when the effective potential V(r) is at a minimum or maximum, i.e. that dV/dr = 0.
Since dV^2/dr = 2 V dv/dr, we can alternatively set the derivative of V^2(r) to zero, as is done in the webpage.
This then gives
L^2 (r-3M) = M r^2
(see the webpage on "circular orbits") or<br />
L = \frac{r}{\sqrt{\frac{r}{M}-3}}<br />
Since d\phi / d\tau = L/r^2 and is constant, the proper time of a complete orbit is given by T, where
\frac{2 pi} {T}= \frac{L}{r^2}
This yields
T = 2 \pi r \sqrt{\frac{r}{M} - 3}
To get the formula quoted, we need to "un-geometrize" the above expression, as the original expressions were in geometric units which assumed that c=G=1.
We see that the expression has units of length - to convert them into time, we divide by c (which is a factor of unity in geometric units).
We could convert M to a length, but I chose to simply re-write the formula in terms of r_s instead. In geometric units, r_s = 2M.
This then gives us, finally
T = \frac{2 \pi r}{c} \sqrt{\frac{2r}{r_s} - 3}
