Why tertiary amines can't show chirality?
The short answer to your question

Under conditions that inhibit inversion, an amine that has three different groups attached is chiral.

The above applies not just to tertiary but also to secondary amines.

Now, why inhibit inversion? Why does it make a difference?
As mentioned by Zhe (and mentioned in Newton's answer) in the comments:

Tertiary amines can definitely be chiral. It's just that they epimerize quickly, but they are definitely chiral when there are 3 different substituents. In addition, you can easily create a tertiary amine that does not invert by tying back the substituents.

If you notice, they are chiral for very short periods of time due to the inversion that takes place in amines. In regards to the time scale – Ammonia ($\ce{NH3}$) flips $4 \times 10^{10}$ times every second. Converting that into the time for one inversion that translates into an inversion every $\pu{2.5 \times10^{-11} s}$ which is $\pu{25 picoseconds}$. Such a scale of time cannot be discerned by any of our current methods to identify stereochemistry.
From http://www.chem.ucalgary.ca/courses/350/Carey5th/Ch07/ch7-3.html


In this case, even though a chirality center is present in each molecule, the sample is optically inactive since the optical activity of the two extremes of inversion
averages out because they are enantiomeric.

For why umbrella inversion occurs, see this answer by SendersReagent

Now, what if we restrict this inversion by attaching free-rotation restricting groups? What happens then?
This restriction brings (or rather removes) a fact into the mix. There is no more inversion possible.  This is because rearrangement into $\mathrm {sp}^2$ cannot take place because of the groups added since it becomes sterically inhibited in becoming planar. An example of this would be quinuclidine which has no way of forming a planar structure.