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I assume by 'without the radio' you mean 'not using commercial RF frequencies' ? OR do you mean without using any form of radio communication whatsoever? In which case, you're left with infra-red Doesn't always mean line of sight, but usually does.
I'm still confused about the 'without the radio' bit. What radio? You mean you would like to transmit your microphone audio to a specfic 'box', that can't be picked up by any other radio (unless they build one themselves). ?
Well depending on your country, and if quality isn't the issue, I would use one of those schematics you see everywhere on the web for 'wireless bug'. Sure, they *usually* transmitter in the 88Mhz to 108Mhz FM band...but why can't you just detune it above or below that? That means no commercial radio could pick it up. I think 109Mhz+ is an aircraft band, but I doubt planes will hear you singing karaoke
As I said, not legal, but pretty simple and the quality can be surprisingly good. For the reciever you would just need a standard commercial FM radio, and modify it so you can tune outside of the normal band.