Upgrading crankset on a 2001 Bianchi Grizzly
I looked at the Bicycle Blue Book page on the 2001 Bianchi Grizzly, and what you included in your question matches up - 110mm spindle BB, Raceface Progidy LP triple crank.

The important thing to get right with a new crank is the chainline. If that's correct the front derailleur range or movement should cover the chainrings properly.

Standard MTB triple chainline was 47.5mm (see this Sheldon Brown Page) and I think it's safe to assume that's what the Bianchi has (you can check it of course). Presumably you also have a rear dropout spacing of 135mm.

You can get a new crank that will fit on the existing bottom bracket, but different model cranks require a different length BB axle to get the right chainline, so you are probably better off replacing the bottom bracket along with the crank. I'm not sure if you have a square taper or one of the splined types, if the later a BB would probably be hard to find. Also, a 18 year old BB probably should be replaced.

If you are going with a Shimano crank, you just need to look up in the Shimano specifications what chainline potential candidates provide (and the intended rear dropout spacing). The Shimano specs typically tell you what BB and axle length you need also In current Shimano products believe only Tourney level cranks are available in 8/9 speed triple and will give you the right chainline, so you might need to look at older used cranks. Fortunately Shimano provides an archive of specification documents.