Jay, it seems the concept here is to just ask people to pray rather than asking someone to "lead" in prayer. If the church is satisfied with this practice, Squire's suggestion would provide a practical solution. Certainly God has no problem hearing all at once. Some believe that Acts 4:24–30, et al., provide a Biblical example for the practice.
It would be interesting to know the historical background of your church and the others of your area that adopt the practice. I believe that this practice, which I have heard referred to as "concert prayer", is rooted in revivalism and the Great Awakenings. Probably churches with the strongest influence from the Separate Baptists (Shubael Stearn, Daniel Marshall, Sandy Creek Assn, etc.) are most likely to retain the practice. Besides in Appalachia, I have found the practice in parts of Tennessee and Kentucky, Southwest Missouri, in north Alabama and north Georgia, especially in "non-cooperative" churches - churches not affiliated with any kind of national body of Baptists. I think westward expansion and movement away from one's "roots" tended to "weed out" this practice. I've seldom seen it in Arkansas, Texas, Louisiana or Oklahoma, even in churches that have retained what might be thought of as other Separate Baptist "peculiarities" - extemporaneous preaching, feet washing, unsalaried ministry, etc, etc..
Some people associate this practice with pentecostalism. But if history is searched, it can be seen that many of the early holiness/pentecostal groups came out of Baptist churches and associations (especially on the east coast) and would have already been practicing "concert prayer" as Baptists.