It seems that the problem can be solved by setting explicit value of the CharacterEncoding global FE option (checked with MMa 8.0.4 and 9.0.0):
SetOptions[$FrontEnd, CharacterEncoding -> "UTF8"];
Export["test.pdf", "кириллический текст"]
An equivalent way (without changing the global FE settings):
Export["test.pdf",
Style["кириллический текст", CharacterEncoding -> "UTF8"]]
Instead of "UTF8" one may set "UTF-8" or "ASCII" with the same effect. The drawback of this approach is that all non-English letters are outlined.
Update
Starting from version 10 (checked with versions 10.4.1 and 11.1.1) Cyrillic text is exported correctly with default setting without converting glyphs into outlines:
Export["test.pdf", "кириллический текст"] // SystemOpen
Here is how exported file looks when opened by Adobe Acrobat 11 (I intentionally selected the first word to show that it isn't outlined):

Moreover, text can be copied from Acrobat and correctly pasted into Notepad.
But importing such PDF as "Plaintext" still fails:
Import["test.pdf", "Plaintext"]
% // ToCharacterCode

{1, 2, 3, 2, 4, 4, 2, 5, 6, 7, 1, 2, 8, 32, 9, 6, 1, 7, 9}
P.S. In versions 10 and 11 CharacterEncoding is not recognized as a valid FrontEnd option.
CharacterEncoding). – Jens May 17 '12 at 06:50ImportString@ExportString["Текст на русском", "PDF"]. No need to useCharacterEncoding. – Szabolcs May 17 '12 at 08:42ExportandExportStringproduce the same data---Exportwill write a readable PDF on my machine (WinXP, Mathematica 8.0.4, like yours). TestExport["test.pdf", "Текст на русском"]again (precisely as I wrote it, don't includeCharacterEncoding), and if it still doesn't give you a readable output (try opening with Adobe Reader, that's what I used), then I have one more guess: set the system language to US English, reboot (just in case), and try again. I know it affects some things, e.g. parsing dates. – Szabolcs May 17 '12 at 09:06