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When I was in school, strictly every abbreviation had a period after each (capitalized) letter, as in U.S.A. These days, it seems ok just to capitalise, as in USA.
Is this a new rule? It does look jarring to have those periods everywhere, and it's annoying to type.
Related: Omitting periods after title abbreviations (Mr, Mrs, Dr).
It depends on the type of abbreviations. USA is an initialism, and as such, does not require periods. The style of acronyms and initialisms no longer requiring periods is now becoming more prevalent.
Abbreviations, generally, still require periods (e.g. Prof. — Oh, and e.g.) Although there are exceptions, especially in British english usage, as noted in your linked related question.
Which abbreviations take periods and which do not is variable depending on the style guide followed.
Lowercase initialisms like e.g., i.e., n.a., and q.v. generally retain their periods to help distinguish them from words (or, in the case of n.a. and relatives, inexplicably take a slash in the middle instead).
My quick-reference book on grammar suggests that the choice may also be (in fact, the author says it is primarily) geographic; i.e. that:
N.B. the above handling of commas was also drawn from the same text (with a bit of tongue-in-cheek).
I have also often seen the periods exchanged for italicized formatting (for example: "eg" instead of "eg" or "e.g."); however, I have not see this convention addressed by a stylist "officially" (I also haven't really looked). That being said, if you find the periods annoying to type, I imagine you'd be even less enthusiastic about toggling italics on and off !
Personally, I try to prioritize clarity and consistency above everything else. This means that I make use of all three conventions, in different contexts; however, once I've chosen a convention for a particular context, I stick to it (this post being an exception; to my delight :D).
The whole thing has something of the same flavour as choosing how to pronounce words imported to English from foreign languages, when one knows how to pronounce the word in that language (e.g. "touché"). If my goal were to communicate clearly, I'd go with the option that I think would maximize the probability of doing so successfully (vs. if the goal were, e.g., to instruct).
