Mine are:
Deftones - My own summer (shove it)
Metallica - For Whom the Bell Tolls
Tool - Stinkfist
Opeth - Black Rose Immortal
Emperor - Alsvaltr/Ye Entrancemperium
Arcturus - Collapse Generation
Black Sabbath – Black Sabbath
Edge of Sanity – Crimson (I feel as if this is cheating, somehow… ;))
Bathory – One Rode to Asa Bay
Metallica - Master of Puppets
Iron Maiden - Number of the Beast
Black Sabbath - Iron Man
This list feels right on to me too. I was leaning towards Sanitarium, but Master is a little more mainstream so probably covers more fans.
Blue Oyster Cult - Veteran of the Psychic Wars (specifically the version from Extraterrestrial Live)
Halford - Silent Screams
Metallica - One
I’d say Iron Man, War Pigs, and Crazy Train are quite possibly the top three. It’s not very imaginative to give a sweep to Black Sabbath/Ozzy, but what can you do?
Master of Puppets and/or Enter Sandman are strong contenders for top-flight metal that moved the genre foreward.
There’s so much that’s great, a “top three” of the entire metal landscape is a virtually impossible goal. There won’t be many nominees that are wrong, just different.
Queen-Stone Cold Crazy
Metallica- So What
Iron Maiden-Run To The Hills
I would replace Bathory with anything by Lizzie Borden
Man, it’s hard to pick just three. I’mm just going to pisck what I think are three really enduring and definitive songs but they aren’t the only three by any means.
Iron Man - Black Sabbath (a genre defining riff)
Hell’s Bells - AC/DC (colossal riff, still sounds kick-ass and spooky, a classic of the genre)
For Whom the Bell Tolls - Metallica (Metallica has to be on this list but it’s tough to pick one definitive song. I think this one really has everything that was ever great about Metallica but I could have just as easily chosen “Master” or “Fade to Black” or “One” or "Seek and Destroy or probably a half dozen others).
Metallica- Damage Inc.
Slayer- Raining Blood
Pantera- Rise
Metallica - One
Sepultura - Ratamahatta
Tool - The Pot (probably a bit premature since it’s a pretty new song but I really like it now)
Just to stretch the definition of “heavy metal” a little:
Clutch - 10001110101
Dream Theater - Pull Me Under
and (for the old headbangers)
Captain Beyond - Raging River Of Fear
(I know these can’t be canon; no umlauts.)
Black Sabbath - Black Sabbath (The first true metal song)
Metallica - One (Simply the most perfect intro solo ever recorded.)
Fear Factory - Martyr (Suffer Bastard Mix) (The begginings of Melo-death and industrial metal)
special mentions -
Morbid Angel - Immortal Rites
Kyuss - Thumb
Killswitch Engage - End Of Heartache
Megadeth - Holy Wars
Las Chicas Son Guerreras - Coz. Not really that much of a metal song, and it was Coz’s only hit, but heck… us girls just like the title (Girls Are Warriors)
Vamos Muy Bien - Obús. Bloody can’t take it out of my head metal song; the Doing All Right lyrics originally apply to being skunk drunk but are often recalled in any kind of SNAFU.
I need one from Barón Rojo or one from Leño… can I just propose all their discography? 
Haha, Spanish girlie metal. Allow me to laugh politely, and add my list, just to spite.
Burzum - Hvis Lyset Tar Oss
Windir - Kampen
Storm - Oppi Fjellet
(Ok, my real list could be One, Hallowed Be Thy Name and omgomgomgomgomg Emerald Sword, but I realize Italian power metal is only a few steps up from Spanish girlie metal)
+2
But no one has mentioned Led Zeppelin yet? At least, I didn’t see it…I would name Zep as metal way before anything AC/DC did - they’re hard rock to me.
For Zep - have to go with Kashmir. May not have crushing guitars like Metallica, et all, but it pounds you with a relentless, yet compelling groove…
I thought about Zep, but I don’t really think they were metal. They were hard rock but metal is …something else. It’s subjective, I guess.
Yeah, but if you go down that path, you’re pretty much forced to conclude that Black Sabbath was the only metal band before the new wave of British heavy metal, and why would they name it that if it were the first? Most of the early metal bands–Deep Purple, Led Zeppelin, Rush, Queen, etc.–would barely be considered metal by today’s standards, but they were definitely thought of as metal at the time.
If you’re going for the three most inluential songs, the list pretty much has to be all Black Sabbath. The songs are “Black Sabbath”, because it was the first; “Paranoid”, because of how often it’s been covered; and “Electric Funeral”, because of the number of genres that were born out of imitations of it.
I had actually disqualified any AC/DC for the same reason I had Zep: By today’s standards, perhaps more “hard rock” than metal overall. I do see the point of keeping things in context, but it opens up a dizzying array of possibilities, making a top 3 list virtually impossible to settle on, at least for me.