I'm a 'no straps' guy, but i'm biased. I liked Hoosegows response on how to deal with it, though.
Although i'm a 'no straps' guy, I do use them for Snatch grip DL's and snatch grip Rack pulls, which is a wide double overhand grip. I don't use them for anything else though. My grip used to be very weak, and used to limit a lot of my lifts. Now, snatch grip variations aside, it doesn't hold anything back - DL's, rack pulls, DB Rows, DB lunges, wieghted pull ups - I actually managed to get my 'towel pull ups' numbers pretty close to my normal pull ups. Doing all these exercises a lone strengthens the grip, but cycling in some thick bar stuff and also what Hoosegow said will see some good improvement.
Also, with DL's, I generally always exhaust my grip on 'double overhand', going until my grip wants to give out, before moving to alternate grip. Same with snatch grip - exhaust it before using straps. I wouldn't recommend that if you were doing a test day but for a general training day it shouldn't effect things too much.
Also, I have tried two different kinds of straps. The classic strap that wraps around the bar and your wrist, and you hold on to it, but I have found that this always slips for me.
Make sure you are looping the straps UNDER the bar, and not OVER the bar. If you're looping over the bar, they won't work very well at all, if you loop under, then I honestly think they'll hold on to anything if you can take the discomfort with them stabbing into your wrists.
p.s Good article on Grip here - http://exrx.net/forum/viewtopic.php?t=5871