In a statement, Amnesty said the "deliberate targeting of civilians" in the bombings that killed at least 16 and injured over 100 Thursday, "shows an utter disregard for human life".
G. Ananthapadmanabhan, Chief Executive of Amnesty International India, said: "The victims of these attacks have a right to justice and reparations. Authorities must conduct a thorough and impartial investigation into the attacks.
"In doing so, they must protect minority communities from the discrimination, violence and harassment they often suffer in the aftermath of an attack."
The bomb attacks took place near Konark Theatre in Dilsukhnagar and near Venkatadri theatre, a few metres away. No individual or group has claimed responsibility yet.
In January 2012, the Andhra Pradesh government awarded compensation to 61 Muslim men who were wrongly implicated in a bomb attack on the Mecca mosque in Hyderabad in 2007.
Many of them had been detained without charge and subjected to torture. No action was taken against those responsible for the torture.
"Shoddy investigations and unlawful police practices can violate the fair trial rights of suspects, and also the victims' right to justice," said Ananthapadmanabhan.
"The authorities must bring those responsible for the attacks to justice in fair trial proceedings in line with international human rights standards, without recourse to the death penalty."