Mr Cash appeared before Judge Cephas Power at Cloverhill District Court via video link from Midlands Prison.
Lisa Cash (18) and her eight-year-old twin brother and sister, Christy and Chelsea Cawley, died after they were attacked and stabbed at their family home in Rossfield Avenue, in the Brookfield area, in the early hours of Sunday morning.
Gardaí from Tallaght station, supported by armed detectives and members of an armed support unit, responded. During a stand-off, officers used non-lethal devices to effect an arrest.
On Monday evening, gardaí charged Andy Cash, also from Rossfield Avenue, with the three murders, and he was remanded in custody by a late special sitting of the district court.
An order was made for him to have a psychological assessment in prison.
As mourners attended the funeral mass of the three victims at St Aidan’s Church in Brookfield on Friday morning, Mr Cash faced his second hearing.
He appeared before Judge Cephas Power at Cloverhill District Court via video link from Midlands Prison.
He did not comment when asked to confirm his name.
Judge Power noted a Garda application to remand the accused in prison further.
Counsel for the defence indicated there was consent.
Judge Power held him in continuing custody to appear again on October 6 for directions from the Director of Public Prosecutions.
During his previous hearing, Garda Rob Whitty gave evidence of arrest, charge and caution. He said he charged the accused with three counts of murder.
"His reply to the charge after caution was 'no comment' to each charge, and he was handed a true copy of each charge sheet," Garda Whitty had said.
The district court cannot grant bail in a murder case which requires a High Court application.