Yes, of course it is possible to transmit without using a fixed frequency. But if you want a receiver somewhere to receive your signal and deliver the audio to a speaker, it has to know what your frequency is at any one moment.
Many phones do as you ask. They transmit audio without using a fixed frequency by using what is called frequency hopping spread spectrum. In this kind of system, the transmitter's frequency jumps around through a range of hundreds or thousands of channels at a high rate of speed. The receiver that is listening to this transmitter happens to know exactly what frequency the transmitter is going to jump to next and it synchronizes so that they jump together to the next channel.
Another kind of system avoids using a fixed frequency by spreading their modulation so wide that it doesn't matter any more what their "frequency" is. This is called Ultra Wide Band modulation. It is theoretically possible to make an ultrawideband transmitter that uses the entire radio spectrum from 1 MHz to 2000 MHz as one big channel and if you do this, you could say that your not using a fixed transmitter frequency.