What kinds of signals are you trying to record? The solutions are different for 'three always-on signals', 'three sources but only one generates a signal at any given time', and 'three sources that are on some times, off others, and might overlap'.
If the signals take turns, you can run them through a multiplexer and reduce the recording end of the problem to a single input. If you need three more or less independent input lines, the easiest solution is also the most expensive one.. throw hardware at it.
A second question would be, "how do you know when to start recording?" For three continuous signals that would be trivial, but for three signals that come on and off at random times it could be challenging.
A third would be, "how do you intend to use the recorded input?" Low-quality audio offers you more wiggle room than high-quality.
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