How about the collection of intervals of the form $a \times [b, b+1)$ where $a$ is any real number and $b$ is any integer? These are half-open vertical intervals that are disjoint and the plane is the union of all intervals of this form.

(Of course, you could turn this around to get intervals of the form $[a,a+1) \times b$ where $a$ is an integer and $b$ is a real number, or any of several other such configurations.)

I may have misunderstood the question.

Start with the collection of half-open intervals of the form $[a,a+1) \times 0$ where $a \geq 0$ is an integer. This decomposes the positive $x$-axis into half-open intervals. Now, for every value of $0 < \theta < 2\pi$, decompose the ray whose angle with the positive $x$-axis is $\theta$ into half-open intervals with the open end of the interval placed at the endpoint nearest the origin.