Here's some suggestions, though I don't know if you mind doubling up on years.

Assassin's Creed:
1- 1191
Revelations- 1511
Freedom Cry- 1735

For the Shadow Hearts games,
Koudelka- 1898
Shadow Hearts 1- 1913
Shadow Hearts Covenant- 1915
Shadow Hearts: FTNW- 1929

Gabriel Knight Trilogy
GK1: 1993
GK2: 1995
GK3: 1999

Tex Murphy: Under A Killing Moon- 2042

Rule of Rose: 1930

Wolfenstein: The New Order -1960

Technobabylon- 2087

Waking Mars- 2097

Wasteland Trilogy:
1- 2087
2- 2102
3- 2107

Yeah I originally conceived this list as one game for every year, and only games that take place within a single year. When there's more than one for a year it's pretty subjective which one I pick, I try to choose the one that I think best represents that particular year.

Appreciate the suggestions and I'll definitely add in the ones that fit!

Gun (1880)
Higurashi (1983)
Vampire the Masquerade (2004)

Thanks @Dadhunter!

Love this list, it's such a cool idea.

I would add a Warhammer 40K game just to have 41st millennium on the list, would be fun.

Got a new one for you; Obsidian, the 1997 adventure game, takes place in 2066.

@ZapRowsdower Thank you!