If a group of people ate together in a large room, such as in a banquet hall, and the person leading the Zimun would not be heard by everybody without the use of a microphone, would it be acceptable for him to use a microphone?
At first glance, this question hinges on the general debate among the Halachic authorities regarding the status of an electronically amplified human voice. Rav Moshe Feinstein (Russia-New York, 1895-1986) was of the opinion that an amplified voice is no different than a normal voice, and one who hears a person’s recitation of a text through a microphone is considered to have heard the recitation directly from the person’s mouth. And thus on Purim, for example, it is acceptable, in the view of Rav Moshe Feinstein, for the reader to read Megilat Ester with a microphone, and all who hear his amplified voice fulfill their obligation no less than they would if he had read the Megila without a microphone. This was also the opinion of the Hazon Ish (Rav Avraham Yeshaya Karelitz, Russia-Israel, 1878-1953).
Hacham Ovadia Yosef, however, disagreed. He felt that when a person’s voice is heard through an amplification system, the sound is not the individual’s actual voice, but rather an artificially produced sound. Therefore, if one reads Megilat Ester over a microphone, the audience does not fulfill their obligation, since they are hearing not the reader’s voice, but rather an electronic reproduction of his voice. Hacham Ovadia allowed using a microphone for the reading of Megilat Ester only in a small room, where the reader’s voice can be heard even without amplification, since the people are indeed hearing his voice. But in a large ballroom, where the voice can be heard only through amplification, the Misva cannot be fulfilled with a microphone, since the people hear a new voice produced by the amplification system, and not the reader’s actual voice.
Seemingly, then, according to Hacham Ovadia, it would not be acceptable for the leader of a Zimun to use a microphone, since the others would not be hearing his voice.
However, Rav Yisrael Bitan (in the English edition of Yalkut Yosef – Berachot) drew a compelling distinction between the case of Megilat Ester and the case of Zimun. When it comes to Megilat Ester, the people need to hear the reader’s voice because of the principle of "Shome’a Ke’oneh" – hearing the recitation of a text can be considered like personally reciting it. We are all obligated to recite Megilat Ester, but we fulfill the obligation by listening to the reading, through which we are considered to have read it ourselves. For this Halachic mechanism to work, we must hear the reader’s actual voice. Since we need to be considered as having personally read the text – which is possible only through the mechanism of "Shome’a Ke’oneh" – we require "Shemi’a," that we directly hear the reader’s voice. When it comes to Zimun, by contrast, there is no such requirement. All that is needed is for the leader to summon the rest of the group to recite Birkat Ha’mazon, and this does not require Halachic "listening." In fact, even if the leader made his announcement in sign language, this would suffice, since all that is needed is for the message to get across to the rest of the group. Therefore, it should be perfectly acceptable for the leader to use a microphone for a Zimun, even according to the view of Hacham Ovadia Yosef, not to mention that this would certainly be acceptable according to the view of Rav Moshe Feinstein. (It should be noted that one of Rav Moshe Feinstein’s considerations in permitting a microphone for the reading of Megilat Ester is the fact that the reading is required Mi’de’rabbanan, and not on the level of Torah obligation, which is true of Zimun, as well.)
There are some authors who report that Hacham Ovadia would not use a microphone for Zimun, but Rabbi Bitan dismisses these reports, in light of the compelling distinction noted above. It is worth mentioning that I clearly recall Hacham Baruch Ben-Haim leading the Zimun in the dining room of Magen David Yeshiva with a microphone. It was a very large room, and it would have been impossible for everyone to hear the Zimun without amplification, and yet Hacham Baruch used a microphone, clearly proving that he felt using a microphone for Zimun is acceptable.
If it is possible for the group to break up into smaller subgroups of ten or more men, so they can make a Zimun without a microphone, this would be preferable, but there is certainly room to allow using a microphone for a Zimun, even though we would not allow the use of a microphone for the reading of Megilat Ester on Purim.
Summary: A microphone may be used for a Zimun so that everyone in the group will be able to hear the leader.