de Broglie came up with an explanation for why the angular momentum might be quantized in the manner Bohr assumed it was. de Broglie realized that if you use the wavelength associated with the electron, and assume that an integral number of wavelengths must fit in the circumference of an orbit, you get the same quantized angular momenta that Bohr did.
The derivation works like this, starting from the idea that the circumference of the circular orbit must be an integral number of wavelengths:
2πr = nλ
Taking the wavelength to be the de Broglie wavelength (λ = h/p), this becomes:
2πr = nh/p
The momentum, p, is simply mv as long as we're talking about non-relativistic speeds, so this becomes:
2πr = nh/mv
Rearranging this a little gives the Bohr relationship:
Ln = mvr = nh/2π