A 3D implementation using the 2D Perlin noise function of Unity3D.

unitythreedimensionalperlinnoise.cs

public static float Noise3D(float x, float y, float z, float frequency, float amplitude, float persistence, int octave, int seed)
	{
		float noise = 0.0f;

		for (int i = 0; i < octave; ++i)
		{
			// Get all permutations of noise for each individual axis
			float noiseXY = Mathf.PerlinNoise(x * frequency + seed, y * frequency + seed) * amplitude;
			float noiseXZ = Mathf.PerlinNoise(x * frequency + seed, z * frequency + seed) * amplitude;
			float noiseYZ = Mathf.PerlinNoise(y * frequency + seed, z * frequency + seed) * amplitude;

			// Reverse of the permutations of noise for each individual axis
			float noiseYX = Mathf.PerlinNoise(y * frequency + seed, x * frequency + seed) * amplitude;
			float noiseZX = Mathf.PerlinNoise(z * frequency + seed, x * frequency + seed) * amplitude;
			float noiseZY = Mathf.PerlinNoise(z * frequency + seed, y * frequency + seed) * amplitude;

			// Use the average of the noise functions
			noise += (noiseXY + noiseXZ + noiseYZ + noiseYX + noiseZX + noiseZY) / 6.0f;

			amplitude *= persistence;
			frequency *= 2.0f;
		}

		// Use the average of all octaves
		return noise / octave;
	}

I deleted the reverse permutations and the noise didn't change much. I still don't understand what it's for. It's fine if you don't remember, it's not too important, the code works just fine.
Thanks for your time!

Glad to hear it was useful after all!