Oh dang...
If we can go back to the PS1 gen, there are quite a few with decent stories. Cliches are hard to avoid though. I definitely agree with your teacher on that point.
Here are a few real quick... (Cliches abound, but that doesn't necessarily detract from all game stories, imo)
Suikoden 3,5
Shadow Hearts 1,2
Grandia Series (Lots of people disagree with me on this one but I feel compelled to list it anyway :P)
Xenosaga
KOTOR 1,2 (Especially #1. Sith Lords kinda left a cliff hanger at the ending but it's still worthy)
Persona 3 (haven't played 4 yet, heard it's better)
Ico SotC (it's a different kind of storytelling, just as epic)
Mass Effect (The story really grabbed me. Loved it.)
MGS series
Lunar SSSC (Heard it's amazing from multiple people, haven't played it yet)
Longest Journey and Dreamfall
Breath of Fire V
Fallout 1,2
Warcraft 3 & Frozen Throne (Wicked story and cinematics)
Beyond Good & Evil
Planescape: Torment (Haven't played it yet but EVERY single person I talk to who owns it praises it's amazing character development and story)
Valkyria Chronicles
Person 1: Does Valkyria Chronicles have trophies?
Person 2: No.
Person 1: Forget it. I'm not buying it.
Person 2: Wait! It's amazing! Unique, charming, drop dead gorgeous... Hello?
