Here's what I want to see:
"I played this game X many times, with Y people, in Z circumstances."
"I think the circumstances in which I played the game had the following effect on play:" (If you do a bunch of stuff in play that's not in the text, talk about that and why you did that stuff. If it's in the text, but maybe not obvious, point it out.)
"I was expecting this from the game, but the experience it gave me was like this"
"The effect of this mechanic on play was to..."
"This aspect of the game was effective in producing this kind of play"
"Games with similar mechanics are the following... ...here's how this game is the same, here's how it's different"
Criticism sandwich! Say something nice about the game at the beginning and the end of the review. Even if you don't really mean it. Even if you have to work hard to think of something. I cannot overstate how effective this technique is in making people actually read and understand things they don't want to hear.
Comparison with other games, or other techniques (in terms of effect on play, not in terms of quality).
Discussion of principles of design.
I care about what the experience of play is like, and how the game's mechanics influence that experience.
Oh! Make the review as short as you reasonably can, without compromising the content.
Here are some things I don't want to see:
A review of the game as a product. I don't care whether you would recommend I buy it.
A biography of the designer. It's cool to put the game in historical context, but it's a review of the game, not of the designer.
Words like "good" or "bad" or "quality". I think words like "useful", "effective" and "taste" are better (and more useful).
A review of the text. I don't really care if the text communicates effectively or not. Let someone else review for that.
Argument from authority. Don't reference theory in the body of the review, unless it's straightforward enough to be explained right there. It's cool to link to "further reading" at the end though. Quotes in the body are iffy. Only do it if it's really the best way to communicate the idea, not to make it sound like some important internet person agrees with you.
Don't make a whole bunch of other essays required reading for your review. Absolutely draw on that knowledge and those ideas, but if it's too complicated to explain in the review, it's probably not that clever anyhow.
Don't affect academic language. The Academic style is a shibboleth for excluding those who haven't paid their dues to the academic hierarchy. Just write like regular. Conversational.
Here are the kinds of games I think should be reviewed:
Any game. All games. Probably your decision about what to review should be informed by how useful your review will be to helping people understand broader concepts of design and play. So, probably a bunch of reviews of near-identical trad games isn't super useful. But I'd be pretty interested in a good review of a few of the more diverse traditional games.