I agree with Hiryuu, SuperUnknown from Soundgarden is intense 
If you’re not a big fan of new Foo Fighters, try their Self-Title; it has a slightly different sound from the rest of their stuff. I figure you already have a lot of Aerosmith, but if you don’t, go get some! Soul Asylum and Collective Soul are nice, too. I’m surprised no one mentioned them. 
Regular old rock… you named more intense stuff than regular old rock. You named the descendants of metal, grunge, and well seether.
Music categories are always hard to do, but I think I found my stuff that comes close to what you are looking for (keep in mind that labels will only limit you, so open your mind and ears to these bands before you condemn based on genre)
-Ozzy Ozzbourne: Blizzard of Oz and Diary of a Madman are the ticket here. Non of that shit he did post-Ozmosis
-Seven Mary Three: The only CD I can think of is American Standard but the sound is right.
-Stabbing Westward: Can’t even think of a CD, but they edge on a more metal sound so don’t worry too much if you can’t find them easily.
-Pantera: Trust me, they rock. Some stuff is rather metal, but most just kicks ass.
-Sepultura: This is a little more metal, but not the cheezy glam metal, the intense kick yo ass metal.
-Filter: Don’t skip this line! Filter’s first CD was badass shit. “Jurrasitol” is the ticket here.
-Toadies: Can’t think of the CD name, but I know that “Possum Kingdom” is a good representation of their sound
-Downset: These guys are like a rock band that wants to rap also. Do We Speak a Dead Language is a good example of their sound. No, they don’t suck, yes they are somewhat moral in thier message, although it comes across with a lot of goodness
-Guns 'N Roses: We’ve all said it by now. This is the ticket. GNR have no comparison.
-Deftones: This does NOT include “Minerva”!!! Deftones are summed up with the song “Bored”
I probably missed completely, but thats some stuff you can at least try out.
BON JOVEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEIIIIIIIIYYYYYYYYYYYAAAAAAAAAEEEEEEEEEEEEE
Violent Femmes, The Cramps, the song “Low Rider” by War, Led Zeppelin (obviously), Mad Season, SOUNDGARDEN (!!!), Alice in Chains, early Judas Priest (for regular rock, later for some great metal), Jethro Tull, The Who, Beach Boys, Blue Oyster Cult, Cream, The Darkness, and while they’re not rock- I’d suggest plain ol’ Darkness, as well (completely different from The Darkness).
Although more glam metally, I highly reccomend Ratt (esp. anything from the album “Invasion of Your Privacy”), Guns N’ Roses, Van Halen, and The Scorpions.
Generic, your tastes kick ass.
You can always go with some of Pearl Jam’s older stuff.
I don’t listen to too much modern rock, since that is what I assume you are looking for, but one really good band that I know of that hasn’t been listed yet is Queens of the Stone Age. Recommended songs are “Better Living Through Chemistry”, “Go With the Flow”, “Feel Good Hit of the Summer”, and “No One Knows”, just to name a few.
Also, even though they’ve been mentioned already, Alice in Chains, because they’re that awesome to be listed that many times. The rest of the bands I would mention would either be named already, or are classic rock, and I don’t know if that’s what you want.
I was thinking about mentioning Collective Soul, but I’m not sure if they’re what Sorc was looking for. Either way, I love 'em. Speaking of which, is their new album any good?
Ooh, somebody mentioned Sepultura. Chaos AD by Sepultura is one of my all-time favorite albums. It’s like thrash metal with a little bit of indigenous Brazillian music thrown in. Awesome stuff.
Chaos AD is an ok album, I like Arise more though.
You’re in the minority. Arise (and Roots for that matter) aren’t bad, they’re just nowhere near Chaos AD’s league. Although if you like generic metal better than the native stuff they implemented into Chaos AD, you might like Arise better.