Do you prefer Shimano, SRAM or Campy?

I start over with my Cross Check all the time. Most recently I put it together as a singlespeed using nothing but parts I had lying around that I took off of other bikes -- a Matrix Aurora wheelset from a 2001 LeMond Nevada City, with cog and spacers from an old Performance singlespeed conversion kit, Ultegra 6500 brifters that I gutted and converted to pure brake levers, an Origin-8 crank, etc. I'm detailing it this way because I think the Cross Check is well suited to this sort of build. At heart, it's a utility bike and so it works well with a utility build. Give it solid components of any sort and you'll be happy. You can buy high end parts, but they won't turn it into a high end bike so I don't think it's worth while.

So, given your specifications, here's what I think I'd get:

Brifters, derailleurs and cassette: Tiagra 4600, a very solid group at a very reasonable price. If you prefer SRAM, substitute Apex. I've got a 12-30 Tiagra cassette, which I like a lot.
Chain: SRAM PC-1050, the PowerLink makes me choose this over Shimano
Crankset: I've been happy with FSA Gossamer cranks, which are often available very cheap. The bottom brackets wear out quickly, but can be replaced with Shimano when that happens. For your purposes, I think 46-36 gearing makes a lot of sense. I use 46-34 because I have a very steep hill to climb daily.
Brakes: This is a quandry. I lean toward Avid SD7's with Travel Agents, because cantilever brakes are almost never trouble free. If you don't mind a tight squeeze over fenders, TRP CX9's are a great option.
Wheels: Tiagra hubs laced to Velocity A23 rims, maybe 32h front and 36h rear.
Handlebars: I love Salsa Bell Lap bars. I might try their Cowbells at some point.
Stem: Profile Design Aris