I used to pronounce the name as muh-RYE-uh.  This is probably because I
have heard a song "They call the wind Mariah!" which uses this
pronounciation.  This song is from the musical "Paint Your Wagon" by
Lerner and Lowe.

This game is based on the books by J. R. R. Tolkien, though, so I got out my
books and looked it up.  Below is the result of my investigation.

-- "The Return of the King", Part Three of the trilogy "The Lord of the Rings"
Appendix F,
   section II: On Translation, paragraph 19
"But Moria is an Elvish name, and given without love; for the Eldar, though
they might at need, in their bitter wars with the Dark Power and his servants,
contrive fortresses underground, were not dwellers in such places of choice."
   section I: The Languages and Peoples of the Third Age,
       subsection Of the Elves, paragraph 2
"Of the Eldarin tongues two are found in this book: the High-elven or Quenya,
and the Grey-elven or Sindarin."
       paragraph 4
"The Exiles, dwelling among the more numerous Grey-elves, had adopted the
Sindarin for daily use; and hence it was the tongue of all those Elves and
Elf-lords that appear in this history."

Appendix E
   section I: Pronunciation of Words and Names
       subsection Consonants
"I    initially before another vowel has the consonantal sound of y in you,
yore in Sindarin only: as in Ioreth, Iarwain."
"R    represents a trilled r in all positions; the sound was not lost before
consonants (as in English part).  The Orcs, and some Dwarves, are said to have
used a back or uvular r, a sound which the Eldar found distasteful."
       subsection Vowels, paragraph 1
"That is, the sounds were approximately those represented by i, e, a, o, u in
English machine, were, father, for, brute, irrespective of quantity."
       subsection Vowels, paragraph 4
"In Sindarin long vowels in stressed monosyllables are marked with the
circumflex, since they tended in such cases to be specially prolonged; so in
du^n compared with Du'nadan." [Note: see ia^ below]
       subsection Stress, paragraph 1
"In words of two syllables it falls in practically all cases on the first
syllable."

-- "The Silmarillion"
Appendix: Elements in Quenya and Sindarin Names
"mor    'dark' in Mordor, Morgoth, Moria, etc."
"ia^    'void, abyss' in Moria"
[Note: ia^ apparently is a single syllable because of the circumflex over
the a.]

Hence, the correct pronounciation appears to be
 MOR-yah (trill the r)

Alternatively, MOR-ee-ah might be acceptable if the ee sound is kept short,
and you don't have to trill the r if you feel uncomfortable doing it.