Why not use Rock, Ice, Ground or Steel Ice Ground? Those are pretty interesting combos.
Or instead of Rock, Flying, Fighting, you could use Ice, Flying, Fighting.
The only main problem with using ice is that
it won't resist whatever it's strong against in the triangle. :/
Precisely. It's a really important trait that the original type trio has, which is why any similar trio with an Ice-type in it wouldn't have the same effect. That includes Rock-Ice-Ground, Steel-Ice-Ground and Fighting-Ice-Flying that you suggested, as well as Rock-Ice-Grass that Mr. Popo suggested. It kinda makes me want Ice-types to have more resistances, because Ice isn't as fragile as the Pokémon franchise deems it to be. I mean come on, huge chunks of ice in real life can be pretty resistant to a lot of stuff.
I've come up with another unmentioned type trio, though. Take Fire-Steel-Rock, which would be an interesting take on choosing to be mainly offensive (Fire-type), part-offensive/part-defensive (Rock-type) or completely defensive (Steel-type) with your starter. It comes with drawbacks, since Fire is a weak type defensively, Rock is a bit strong on both ends but could use more of a niche (kind of like a jack-of-all-trades, master of none thing), and Steel only hits two types super effectively, and is a pathetic offensive STAB. If you've played other RPGs that let you choose between something like that, then it might look like a familiar mechanic. It would be a fun way to implement that with Pokémon. But, you guys don't like trios with one of the types in the original trio, right? Shame. Still, it's a nice balance, if I do say so myself.
This is all just conjecture, though, so don't take my word for it.