It is not necessary that every time during DikshA, the Guru must speak the Mantra onto the disciple's right ear.
The rule, as to which ear, depends on the Varna of the disciple and the gender. The rules of DikshAs also can vary with SampradAyas, ParamparAs and Mantras themselves.
ViswaSAra Tantram, RAdhA Tantram etc , for example, mention speaking Mantra unto right ear (Daksha Karna)
And, i know about Mantras, which are to be spoken into the ears not for 3 times but 8 times.
From Vashishta Tantram (a Vaishnava scripture):
Tatah RishiAdisamyuktam Mantram Gururdakshina karne Trih SrAvayitvA
VAma karne Sakrit SrAvayet.
Thereafter the Guru should speak the Mantra, along with Rishi, Devata
and other Angas, 3 times in the disciple's right ear and 1 time in the
left (vAma karne Sakrit).
From the Gautamiya Tantram (another Vaishnava Text):
Nyasa jalam tasya dehe Guruh samnyasya jatnatatah |
Daksha karne Vaden
Mantram TrivAram PurnamAnasaha ||
Dakshe Iti DvijAtivishayam..
The Guru should do Nyasas with care on the disciple's body and speak
mantra into his right ear three times. This rule of speaking unto
the right ear is only for the DvijAs.
When the disciple belongs to one of the Dvija Varnas then, the Mantra is spoken unto the right ear first for 3 times, thereafter 1 time it is spoken unto the left ear.. Now, this process is reversed when the disciple is a Sudra or a woman.
From Bhairavi Tantram (a ShAkta Tantra):
Daksha Karne Triso VidyAm EkorchArena ChocchAret |
DvijAtinAm Stri SudrAnancha VAmatah ||
The Guru shall speak the mantra into the disciple's right ear 3 times
and into his left ear 1 time. This is the rule for the Dvijas, for
women and Sudras it's just the opposite i.e 3 times in the left ear
and 1 time in the right.
The same repeats the Rudra YAmala Tantram (which is another ShAkta scripture):
Gurustu PrAngukhobhutva ShishyA Prachi Mukha Stithah |
Dakshine Karne VAme chaiva TathA Sakrit ||
Viparitam Tato Geyam
StreesudrAnancha VAmatah |
Guru should face the east and the disciple the west direction. Guru
should then speak the mantra into the disciple's right ear 3 times and left
ear 1 time. For Sudras and women the process will be reversed.
And, besides the scriptures i have mentioned, there are plenty of other Tantras and Agamas, where such rules of DikshAs are detailed.