I use a grease that is used on boat trailer wheels, and
it seems to do the job on my 10301. Remember, don't
put more that a pump, or two in each wheel.

Does the 10301 have the aluminum-cased clutch/gear pack with grease fitting at each rear wheel? Or, do you have to remove the wheel to get to the drive gear which has a ratchet built into it? If it has the grease fitting, lbpod is right - two pumps max. You need to use NLGI #2 or lighter, down to #0. If you read the cartridge you have in your grease gun, it probably is a 'lithium complex' grease anyway. I like to use white lithium grease on them, but I don't have any mowers of that type anymore. On mine I have to take the wheel off, which is not wasted because I clean and grease the wheel gear and axle at that time, to get to the ratchet gear. I use mineral sprits and a toothbrush to clean the gear, then shoot it full of spray white lithium grease using the center hole in the axle. You can also use a grease gun there if you have the rubber-tipped 'point' accessory.
I just did the rear end on one of my mowers the other day, and I used some grease I've had around for awhile, 'Slick 50' #2 grease. This stuff is green in color but has a nice consistency for manual greasing of axles, gears, etc. I think basically any #2 or lighter grease you find at an auto supply will work. Manufacturers state the type they use because it is known that mixing grease types is not good in high-pressure applications like vehicle wheel bearings and ball joints. For a mower axle or clutch housing, it's probably not very important. Mainly you don't want to over-service that housing because it will make the wheels drag on pull-back.

gbe: Either lithium or a quality Moly grease will suffice within this application. As recommended do not mix greases whenever possible . Over pressure greasing will cause the aforementioned hyd lock potential .