I moved recently, from a city who's water I had grown up on, to a city with a reputation for bad water. The first day, I picked up a DIY reverse-osmosis water filter system at my local big box hardware store for 130 dollars, installed it (and a new faucet, to make room for the water spigot) in an evening. The ice cubes from that water are crystal clear, and the water tastes fantastic (rather, doesn't taste like anything). Sediment and carbon filters will need to be replaced perodically (4 mos), but the RO membrane will last for 24-36 mos, and is only $50 or so.

It was a bit of an up-front cost, but forget Brita pitchers or faucet mount gadgets, this is now the only way I'll filter water for ice, drinking, and cooking.