Why doesn't sampling a periodic continuous-time signal yield a periodic discrete-time signal?
If the ratio between your sampling frequency and the frequency of your signal is irrational, you will not have a periodic discrete signal.

Assuming you have a 1-kHz sine wave and you sample at 3000*sqrt(2) Hz. You will have approximately 4.2 samples per period. However you will not be able to sample the sine wave exactly at the same place. Hence your digital signal will not be periodic. 

However, if you sampled the same 1-kHz signal at 4 kHz, you would get a periodic discrete signal. The period would be 4 samples.