Dumbbells hanging at side. It can also be good grip training as well. Initially, your grip may be weak, but it will catch up in strength quickly. You could always try straps. I generally hate the use of straps, because I think that there's no point in being strong if you can't transfer that strength through your grip into something in the everyday world. Maybe you could use straps at the end of your walking lunge workout AFTER your grip is already exhausted.
Have you tried a hook grip? This is a stronger grip for me. To do it, you wrap your thumb around the bar first, then your fingers go over the thumb. This kind of grip allows me to hold most weight a lot longer, but I still haven't been able to hold REALLY high weight with it because it just hurts my thumb too damn much. Some people Deadlift with it and even do Olympic lifts. I couldn't imagine doing a dynamic grip like an OL with it. I think I'd break my thumbs.
I found a link to a hook grip description:
on the left side of the page, scroll down to "hook grip" and click on it for a description and picture.
I tend to use three fingers to cover my thumb instead of two. Maybe that hurts more. I ought to try two.