I had persistent shin splints for a number of years. Shoe-wise, my advice is to find a shoe that fits your foot well and replace the insoles with an orthotic. It doesn't need to be a custom-made orthotic - I have had a pair of those, but I find Superfeet suit me equally well.
Whilst the shoes helped, foam rolling is what got me pain free. I use a solid foam version for the shins - I also own a trigger point roller which has a plastic tube as the inner, but find it too firm for the shins. A couple of minutes a day is all I need, though be warned... over time, lack of leg hair where you're rolling can be a side effect!
(Asics Gel Resolution have been my shoe of choice for around a decade in case you're curious.)
Agreed with Dags, I think rolling out the lower leg muscles is really the only thing that helped. Prior to playing, I would stand on a tennis ball one foot at a time massaging and loosening up the bottom of the foot and tendons, then on my knees I would lean forward and use the roller on the front muscle (extensor digitorum longus) using my body weight as leverage to really stretch the muscle out. Then switch legs then move to the claves by sitting on my butt, placing the roller under my calves, lifting up my body and using my body weight to roll out the calves. The goal to keep them stretched, limber, and loose.
Similar to this guy although I would 'activate' the bottom of the feet prior to rolling:
BUT I also never got shin splints with the Nike CB 4.3 or the Lunar Ballistec 1 and 1.5 so I've only played in them since they debuted and stockpiled a ton of them cause I was paranoid to switch shoes. I could usually get away with a Vapor or Cage for a single match without inflicting any shin splint pain, but if I played multiple days in a row with those I would start feeling it. I can play every day with the LBs.
After a year or so of the PT described above and consistently using the LBs I've not had any issues.