I made a vow to fast three months but I do not remember if I intended to fast them consecutively or not. I was hesitant about this matter at the time of making the vow and now I do not remember what my final decision was. Am I required to fast them consecutively?

All perfect praise be to Allaah, the Lord of the worlds. I testify that there is none worthy of worship except Allaah and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and messenger.

Since you are doubtful whether you made the vow to fast the three months consecutively or not, you are not required to observe consecutiveness. The original ruling is that you are not liable to do so. However, it is more cautious to observe consecutiveness because the Prophet, sallallaahu ‘alayhi wa sallam, said: "Leave that which makes you doubtful for that which does not make you doubtful." [Ahmad]

You should also know that making vows is disliked in the Sharee'ah. It was narrated on the authority of Ibn ‘Umar that he said: "The Messenger of Allaah, sallallaahu ‘alayhi wa sallam, forbade vows and said, 'It (vow) has no effect against fate but is a means whereby (something) is extracted from the miserly person.'" [Al-Bukhari and Muslim]

However, when a person makes a vow to do an act of obedience, then he is required to fulfill his vow. The Prophet, sallallaahu ‘alayhi wa sallam, said: "Whoever makes a vow to obey Allaah should obey Him." [Al-Bukhari]

Allaah Knows best.