Du'a can be accepted every time as far as we sincerely seek from Allah with the Waseela (mean) of His most beloved servants. Allah (SWT) Says in Quran : Seek from Me, I will respond to your supplications. This means that Du'a is accepted when ever you seek from Allah. Nevertheless, there are many special times where Du'a is more recommended like:
1. During Sojood (Prostration) because you will be very near to Allah while you are in prostration.
2. Between Azaan and Iqama.
3. Before sunrise and before sunset.
4. After every Salah (Prayer).
5. On the last sixth part of the night (Sahar).
6. On Laylatul Jumu'ah which is the the night of Thursday.
7. On midday of Friday and before sunset of Friday.
8. When rain starts falling.
9. During and after Ziyarat of Ahlul Bayt (AS) especially after Ziyarat Ashura which is highly recommended every day.
10. During Qunoot of Witr which is the last Rak'ah of Night Prayer.
'And many other times.