The simplest one-IC transmitter is this: Wire five of the gates of a 74HC14 hex schmitt inverter in series, with a 1MHz crystal completing the loop. Connect the input of the “spare” gate to one of the intermediate outputs and connect a 220 ohm resistor from the output of this gate to 0V. Connect the antenna to the junction of the gate output and the resistor. Connect a carbon microphone (from an old telephone handset) in series with the supply (4 * 1. 2V AA cells in series).

Modulation comes from the fact that the mic varies the voltage available to the oscillator (but the crystal oscillator keeps running at a constant frequency). The 220 ohm sink resistor is there to ensure that there is a severe current gradient everytime the oscillator switches.

Note: you will not be able to pick this up on a set with a digital tuner, because it broadcasts on 1MHz and digital tuners only go up in multiples of 9kHz.