"do ... while" loop equivalent in Mathematica

While While[procedure; test] works, it looks very similar to While[test, procedure]. The only difference is ; vs ,. While is not the most commonly used construct, so when used like this there's a high chance of misunderstanding/misreading.

If readability/reliability is a concern (for example a collaboratively developed published package), I'd use the longer but clearer

While[True,
  procedure;
  If[Not[test], Break[]]
]

The only argument here is readability and "defensive programming" (extra effort to avoid accidental problems). Readability is subjective. If you are the only person who writes/reads the code and you get used to this use of While (and thus always pay special attention to the , vs ;) then this argument doesn't apply.

Why not write your own and place it in your init.m file?

SetAttributes[DoWhile, HoldAll];

DoWhile[procedure_, test_] := While[procedure; test]

Here's another way to write your own:

SetAttributes[doWhile, HoldAll];    
doWhile[expr_, test_] := CompoundExpression[expr, While[test, expr]]

Or just:

doWhile[expr_, test_] := (expr; While[test, expr])