We don't know which writing system was used by the orcs in Middle-Earth. The Ring Inscription was done with Tengwar.

According to Appendix E of LOTR III:

The Cirth in their older and simpler form spread eastward in the Second Age, and became known to many peoples, to Men and Dwarves, and even to Orcs, all of whom altered them to suit their purposes and according to their skill or lack of it.

However in LOTR II (Book Three, Chapter I: The Departure of Boromir) Legolas says that Sauron does not uses Elvish runes:

on the front of their iron helms was set an S-rune, wrought of some white metal.
‘I have not seen these tokens before,’ said Aragorn. ‘What do they mean?’
‘S is for Sauron,’ said Gimli. ‘That is easy to read.’
‘Nay!’ said Legolas. ‘Sauron does not use the elf-runes.’
‘Neither does he use his right name, nor permit it to be spelt or spoken,’ said Aragorn. ‘And he does not use white. The Orcs in the service of Barad-dûr use the sign of the Red Eye.’ He stood for a moment in thought. ‘S is for Saruman, I guess,’ he said at length.

So, probably Cirth was used only by Westron-speaking orcs of Misty Mountains and Moria. And it seems Mordor orcs used some hieroglyph-like symbols (at least we sure about Red Eye).

But many fans tried to invent writing systems unique for their dialects. This page will list some of them.

Za Maush (SV “letters”) is “abugida” writing system invented in late 1990s - early 2000s for Svartiska

“Mordor Alphabet” was invented by The Second Nazgul for “Dark Messages” forum board approximately in 2002. I have saved only part of it. There was also a font by Json Aptenos, also lost.

This alphabet was also born in “Dark Messages” board by request of The Land of Shadow site's administrator in 2003. Letters were invented by Sarumans Granddaughter and font file was created by Lugrekh. It was based on the cirth modified for writing with ink or paintbrush. It has the flaw, that cirth's runes for “kh” and “g” were swapped during creation of this alphabet (compare final table with version 1).

See Alphabet section of BlackSpeech.ru site for detailed description.

Maushur (in Svartiska) or Mâshur (in Zhâburi B) is basically a Tengwar stylized to cuneiform. It was invented by Björn Axén before 2011.

Svartiska and Zhâburi versions differ in depicting numbers:

Kirkhi (or kirkh) is the runic alphabet based on Mâshur which was made for Zhâburi dialect.

Angerthas Mordor is variant of Cirth (Angerthas Mordor) invented for Zhâburi dialect.

Runes within brackets ( ) are not used in the Mordor version. Runes with a little X have a different sound value than in the Moria runes and the original sound is written within brackets. Two new runes for o and ô were created, but original runes (50 and 51) also may be used.

Daniel Reeve invented Orkish writing system for The Hobbit movie trilogy, which he calls “goblin lettering”. It can be found in the bounty offer for Thorin. The image is taken from Elendilion website which also presents some examples.