To find the equation of the diameter of the circle, we need to analyze the given equation and determine its center and radius.

1. Identifying the center of the circle:
The general equation of a circle is given by (x - h)^2 + (y - k)^2 = r^2, where (h, k) represents the center of the circle and r represents the radius.

In our given equation, x^2 + y^2 - 6x - 2y = 0, we can rewrite it as (x^2 - 6x) + (y^2 - 2y) = 0. Completing the square for both x and y, we get (x - 3)^2 - 9 + (y - 1)^2 - 1 = 0. Simplifying further, we have (x - 3)^2 + (y - 1)^2 = 10.

Comparing this with the general equation, we can see that the center of the circle is (3, 1).

2. Finding the radius of the circle:
The radius of the circle can be found by taking the square root of the constant term in the equation. In this case, the constant term is 10, so the radius is sqrt(10).

3. Determining the equation of the diameter:
A diameter of a circle passes through the center and divides the circle into two equal halves. Since the given circle passes through the origin, we know that the diameter must pass through the origin and the center of the circle (3, 1).

Using the two points (0, 0) and (3, 1), we can find the equation of the line passing through them using the slope-intercept form, y = mx + b.

The slope of the line passing through these two points is given by (1 - 0)/(3 - 0) = 1/3.

Substituting one of the points into the equation, we have 0 = (1/3)(0) + b, which gives us b = 0.

Therefore, the equation of the diameter passing through the origin and the center of the circle is y = (1/3)x.

4. Matching the options:
Comparing the equation of the diameter obtained, y = (1/3)x, with the given options:
a) x - 3y = 0 (Incorrect)
b) x + 3y = 0 (Incorrect)
c) 3x + y = 0 (Incorrect)
d) 3x - y = 0 (Correct)

Thus, the correct answer is option D, 3x - y = 0.