What training is recommended to climb Mt. Whitney?

I agree with George Anders that your long term training should focus on stamina. However, immediately before your Whitney climb, altitude acclimatization is all-important. Do not assume that you will have no problems with altitude sickness on Whitney just because you can hike and climb well at 10,000 feet. For those coming up rapidly from near sea level, there is an enormous difference between exertion at 10,000 feet and at 14,500 feet. I recommend camping and hiking for the previous two to three days near or above 10,000 feet to acclimatize before your summit days. Possible locations include Onion Valley and the Tioga Pass area. The campsites at Saddlebag Lake are at 10,100 feet and there are many training hikes in the area, so that's a great acclimatization area for Northern Californians.  A moderately fit person who has adapted to high altitudes for a few days will often do better on Whitney than a super athlete who tries to zoom from sea level to the summit in 36 hours.

I hiked Mt. Whitney in 11 hours car to car, which is on the faster side of typical. I hike a lot, but the training that really boosted my hiking speed was running. I used the Couch 2 5k phone app which starts from near zero running ability, because I found as an avid hiker I could only run for a few minutes. From there, my endurance increased dramatically. I could hike much faster with fewer breaks and less water. I also did a lot of light weight squats to strengthen my quads for the walk down. Reaching the summit is only half of the distance, after all. If you can, hike Mt. Langley from Horseshoe Meadows a few days before to acclimatize. If you sleep at the trailhead, hike the next morning, and return in the evening, you will have spent about 24 hours at elevations 10,000-14,000'. Most people sleep at the Whitney Portal trailhead (8,360') as well. An acclimatization hike from the Portal to Lone Pine Lake (10,000') is possible and requires no permit.

Building on Jim's comments, there's nothing better than long hikes at increasing altitude in the weeks before your Whitney hike. Last year, the week before we climbed the mountaineer's route, we backpacked into McGee Creek and climbed a 13k peak, then the day before Whitney we hiked to Kearsarge Pass. Didn't do a lot of training before these hikes. Whitney was a relative breeze.