The reason for the appearance of Pi in the Maxwell-Boltzmann distribution can be rationalized by the following steps:
1) The probability of a given state in a thermal ensemble is proportional to the Boltzmann factor $\exp{(-E/kT)}$.
2) The energy of the point gas particle is expressed by its momentum $p$, the Boltzmann factor is then $\exp{(-p^2/2mkT)}$.
So we know that the probability of a molecule having a total momentum of $p$ is proportonal to $\exp{(-p^2/2mkT)}$. To make a proper distribution $f_p$ that tells you the probablity of a molecule having a momentum between $p$ and $p+\delta p$ we must make sure that the integral of $f_p$ gives 1. That means we got to normalize it by the integral of the Boltzmann factor:
$ \int \int \int_{-\infty}^{\infty} \mathrm{d}p_x \mathrm{d}p_y \mathrm{d}p_z \exp{(-(p_x^2+p_y^2+p_z^2)/2mkT)} $
Now, know that this integral is the famous Gaussian integral:
$ \int_{-\infty}^{\infty} \exp{(-x^2)} \mathrm{d}x = \sqrt{\pi} $
Which, when applied for the three spatial dimensions, gets you the $\pi ^{-3/2} $ factor that you must have encountered in your lecture somewhere.
Side note: $\pi$ always appears when you are dealing with 3D problems simply because you almost always 'partition' the space into small spheres and you get the volume of the sphere in your results.