Attorneys for John Allen Rubio argued jurors at his trial in Cameron County failed to consider Rubio's insanity when they decided he should die for killing the children -- ages 3, 1 and 2 months -- in 2003 because he considered them possessed. Appeals also challenged whether the Rio Grande Valley man would be a future danger and whether jurors improperly considered some evidence during his punishment trial.
The court Wednesday rejected the arguments. He was convicted at a second trial three years after a previous capital murder conviction was overturned.
He doesn't have an execution date and still could appeal to the federal courts.