I started with an H-Bar on my singlespeed, then bought a second for a geared bike.
I tried X0 grips, X0 triggers, and Paul thumbshifters with a Jones H-bar on my Turner Flux. All of them worked (the thumbshifters were the best of the lot), though compromised the multi-hand position that makes the Jones bar so nice.
I spent a few months using the thumbshifters until finally switching to a Mary bar with gripshifts. As much as I wanted to use the Jones bars (the bar, along with the X0 triggers and Paul's are in a spare parts box now, doh!), the thumbshifters just didn't cut it full-time. If I were to try again (and I doubt I will, I now have Mary bars on all my geared bikes), I think the only practical option is the integrated Shimano levers ... I'm just not keen on buying yet another set of shifters.