I used to have a problem with this. Three years ago I did a 10-day tour down the Oregon coast. My left hand went numb and it lasted for a month after the tour was over.

Here's what I did:

1. My handlebars used to be lower than my saddle. I raised them until they're level. That puts less pressure on your hands. If you have a quill stem, check out Nitto Technomic stems. If you have a threadless headset, check out the BOA stem from Profile Design.
2. I re-wrapped my bars. First I put some gel pads on the bars themselves - I've used Aztek gel pads and Specialized and they both work. Then I wrapped the bars with the foam tape that came with the gel pads. Then, over that, I wrapped two layers of cloth tape. My bars are pretty fat, but I have big hands so that doesn't bother me. Not only is there a lot of cushioning, but I think the larger diameter grips spread the weight of my upper body over more surface area on my hands.
3. I use gel gloves.
4. I use drop bars. They give me more options for hand placement. I change positions often. My hands tell me when; I don't even think about it. I wouldn't tour with straight bars (like on mountain bikes.) I guess if you don't have a problem with numbness, they're okay, especially with comfy grips and bar ends, but I like drops better. Trekking bars are also an option.

I just finished a 500 mile tour with no numbness. Of course, there's always a little bit of residual effect from being on the bars all day. I find that when I write in my journal after the ride, my handwriting is messy because my hands feel a little "funny", but it's not to the point of numbness.