A few of the "classic" Helsinki eateries with Finnish cuisine:

Savoy - Back to (refined) basics on a constantly high level. An August crayfish dinner can be spectacular.

Lehtovaara - A bit out of the centre. Best bet is the Sunday brunch/lunch buffet.

Sea Horse - Could be found on Manhattan. Urban Helsinki at it's best with workman's fare.

Salve - Ditto, with a maritime twist.

Kosmos - Traditional artists haunt, nice mix of Russian, French and domestic influences on the menu.

Elite - Similar to Kosmos, but with an excellent, romantic summer terrace.

Personally I'm not a big fan of Lapland cuisine but would guess the showcase places in Helsinki are at least acceptable.