The RC oscillator is used because it is cheap. Here is the internal circuit from the PIC16F54 datasheet:
The RC charges until the voltage exceeds the upper threshold of the buffer, at which point the transistor switches on and discharges the capacitor to the lower threshold of the Schmitt trigger buffer, and the process repeats after the transistor turns off. The net result is a sawtooth-like waveform across the capacitor. The transistor is much lower 'resistance' than the external resistor so the discharge is very abrupt relative to the charge.
If you really want to use an LC circuit, you can use the crystal/resonator oscillator options, but most engineers would pick a ceramic resonator or a crystal where a bit better stability than the RC can offer is required.
Newer PIC chips have a calibrated internal RC oscillator that is attractive in many useful applications- some are now good enough to allow reliable serial communications over an acceptably wide temperature range. The next step up in common requirements is timekeeping and that requires a crystal unless you're willing to sync with some external clock very frequently.