Why can't I place a node under another node?

Have a look at the TikZ manual "17.5.3 Advanced Placement Options". The keys below of, above of etc, have been deprecated in favour of the more flexible below, above etc that are available with the positioning tikz library.

As an example, you can write [below=of A_3] and even [below=2 of A_3] to specify a distance.

Note that, unless you specify the on grid option, the distance between the nodes will be the length of the space between them, not the distance between their centers.

EDIT: After reading the comments I realised the problem may be a misinterpretation of the effect of the option below: the effect of below=of A is to place the node below A with a distance between their edges of 1 (the exact measure of which depends on scaling etc but defaults to 1cm). Now, it seems you are expecting dry to be placed below A but very close to it. That is what would have happened with a label=below:dry style: \node[state] (A_3) [right=of A_2,label={[state]below:dry}] {Sunny};. You can also specify a distance, try below=.1 of A. In case this is not the problem, try being more specific and report exactly the nature of the issue (i.e. what does "does not like the usage of below" mean? Do you get an error?)

Given the position of the red line, you clearly want

below of=C_1

rather than

below of=A_3

Otherwise, the node will be placed below A_3 i.e. on top of B_3.

You seem to be under the impression that your nodes are arranged like this:

A_1 B_1 C_1
A_2 B_2 C_2
A_3 B_3 C_3

but you have drawn them in this way:

A_1 A_2 A_3
B_1 B_2 B_3
C_1 C_2 C_3